{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div align=\"right\" style=\"text-align: right\"><i>Peter Norvig<br>Updated Sep 2018, May 2020</i></div>\n",
    "\n",
    "![the count](https://vignette.wikia.nocookie.net/muppet/images/9/90/CountCountsLP%282%29.jpg/revision/latest/scale-to-width-down/280?cb=20140628202329)\n",
    "\n",
    "# How to Count Things\n",
    "\n",
    "Sesame Street teaches us to  count up to ten using our fingers. A computer doesn't have fingers, but it too can use brute force, enumerating things one by one, to easily count up to a billion or so. So in that sense, a billion is a small number. It is rare to get more than a billion <i>things</i> together in one place, but it is common to encounter situations where there many billions of <i>combinations</i> of things; indeed many problems have <a href=\"http://norvig.com/atoms.html\">more combinations of things than the number of atoms in the Universe</a>. \n",
    "\n",
    "Thus, for really big numbers we need a portfolio of counting strategies. Here is a partial list:\n",
    "\n",
    "- **Brute Force enumeration**: Generate all the things, and count them one by one. <br>*Example*: How many odd numbers from 1 to 10? Answer: Generate {1, 3, 5, 7, 9}, count 5.\n",
    "- **Enumerate and test**: Generate a larger set of candidate things, and count the ones that satisfy some criteria. <br>*Example*: How many odd prime numbers from 1 to 10? Answer: Generate {1, 3, 5, 7, 9}, test each one for primality, count 3.\n",
    "- **Incremental enumeration**: When the things we are counting have parts, don't generate all possible complete things and then check each one to see if it is valid. Instead, generate the first part of a thing, and check if that part is valid so far. If it is, generate all the possibilities for the next part, but if it is not, stop there. That means that many invalid things will never get generated, saving time.<br>*Example*: Given a set of *n* cities, how many acyclic paths through some subset of cities are in alphabetical order by city name? Brute force enumeration would generate all permutations up to length *n* and check if each one was alphabetical; incremental enumeration would start generating paths and only extend them with cities in the correct order.\n",
    "- **Abstract enumeration**: put the things into equivalence classes, and calculate how many there are in each class. Then we don't have to enumerate all the things; we can consider many at the same time.\n",
    "- **Divide and conquer**: Split the problem into parts, solve each part, and combine the results.<br>*Example*: How many ways are there of getting a straight flush in poker? We can divide the problem into 4 subproblems, one for each suit. Then for each suit we can say: a straight can have one of 10 different high cards (5 through Ace), so there are 10 possible straights for each suit. The total number of straight flushes is 40, which you can think of either as multiplying 4 and 10, the numbers from the two independent components of the problem, or you can think of as adding 10+10+10+10 for the four disjoint parts of the problem.\n",
    "- **Recursive divide and conquer**: often we break a problem down into smaller pieces that are recursive instances of the same type of problem. We conquer by solving the smaller pieces and combining results.<br>*Example*: How many permutations are there of *n* things? If *n* > 0, solve by finding the number of permutations of *n* - 1 things, and multiplying by *n*.\n",
    "- **Formula calculation**: Use mathematical thinking to derive a formula for the number of things. <br>*Example*: How many odd numbers from 1 to *n*? The formula is ⌈n/2⌉, meaning \"divide *n* by 2 and round up\".\n",
    "- **Remembering**: Sometimes there are multiple ways to break a problem into subproblems, and when solving the big problem we may come across the same subproblem more than once. We can remember the solution to the subproblem so that we don't recompute it multiple times. We call that *memoizing* or *caching* the results; we can use the decorator `@lru_cache`.\n",
    "- **Simulation**: Sometimes it is difficult to exactly count all the things. But you can do a random simulation in which you record the things that randomly come up, and use those results as an estimate.\n",
    "- **Visualization**: When you're stuck, making a chart or table or plot of examples can help you see patterns that can lead to solutions.\n",
    "- **Checking**: make sure that your calculations work for small test cases that you verify by hand. Create two different programs and check that they agree with each other. A common approach is to have one straightforward but inefficient program that is easy to see is correct, and one program that is optimized for speed but more complex. If the two programs agree on the small inputs, you can have more confidence they are both correct.)\n",
    "- **Standing on shoulders**: it is fun to solve a problem on your own, but sometimes the right approach is to look up how others have solved it before. Sometimes you need to do some work to understand the problem better before you know how to search for a prior solution.<br>*Example*: In how many ways can a convex polygon be cut into triangles by connecting vertices? You could type the question directly as [a search](https://www.google.com/search?q=In+how+many+ways+can+a+convex+polygon+be+cut+into+triangles+by+connecting+vertices) and find some helpful answers. You could also solve the problem for small polygons (with *n* = 3 to 7 sides), note that the sequence of answers is [\"1, 2, 5, 14, 42\"](https://www.google.com/search?q=%221%2C+2%2C+5%2C+14%2C+42%22&oq=%221%2C+2%2C+5%2C+14%2C+42%22) and see that a search reveals that  these are called [Catalan Numbers](https://en.wikipedia.org/wiki/Catalan_number) and that there is a lot written about them.<br>*Example*: A coach wants to create a basketball lineup of three shooters and two rebounders. She has seven shooters and five rebounders to select from. How many different lineups can she make? Entering the text of the whole problem as a search query gives [results about basketball](https://www.google.com/search?q=A+coach+wants+to+create+a+basketball+lineup+of+three+shooters+and+two+rebounders.+She+has+seven+shooters+and+five+rebounders+to+select+from.+How+many+different+lineups+can+she+make%3F&oq=A+coach+wants+to+create+a+basketball+lineup+of+three+shooters+and+two+rebounders.+She+has+seven+shooters+and+five+rebounders+to+select+from.+How+many+different+lineups+can+she+make), not about combinatorics.  You need to understand the problem and standard terminology well enough to realize that a better query is [7 choose 3 * 5 choose 2](https://www.google.com/search?q=7+choose+3+*+5+choose+2).\n",
    "\n",
    "# Preliminaries: Imports and Utilities\n",
    "\n",
    "Before getting started, here are the necessary imports, and four oft-used utility functions:\n",
    "- `quantify` (from [the `itertools` module recipes](https://docs.python.org/3/library/itertools.html)) takes a collection of things and a predicate and counts for how many things the predicate is true. It is designed for the **enumerate and test** strategy, but can be used for the **brute force enumeration** strategy by omitting the optional predicate (as long as none of the things you want to count are false (like `0` or the empty string).\n",
    "- `total` totals up all the values in a `dict` or `Counter`. Helpful because `sum(counter)` would sum the keys, not the values.\n",
    "- `iterate`: The pattern of repeatedly calling a function *n* times will be common in this notebook; `iterate(f, x, n)`, a function [borrowed](https://hackage.haskell.org/package/base-4.14.0.0/docs/Prelude.html#v:iterate) from  Haskell, encapsulates the pattern.\n",
    "- `same`: Tests if two functions return the same output for all the given inputs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import random\n",
    "from collections import Counter, namedtuple\n",
    "from functools   import lru_cache\n",
    "from itertools   import product, permutations, combinations, islice\n",
    "from math        import factorial, log\n",
    "from statistics  import mean, stdev\n",
    "from sys         import maxsize\n",
    "\n",
    "def quantify(iterable, predicate=bool) -> int: \n",
    "    \"\"\"Count the number of items in iterable for which the predicate is true.\"\"\"\n",
    "    return sum(1 for item in iterable if predicate(item)) \n",
    "\n",
    "def total(counter) -> int:\n",
    "    \"\"\"The sum of all the values in a Counter (or dict or other mapping).\"\"\"\n",
    "    return sum(counter.values())\n",
    "\n",
    "def iterate(f, x, n) -> object:\n",
    "    \"\"\"Return f^n(x); for example, iterate(f, x, 3) == f(f(f(x))).\"\"\"\n",
    "    for _ in range(n):\n",
    "        x = f(x)\n",
    "    return x\n",
    "\n",
    "def same(fn1, fn2, inputs) -> bool:\n",
    "    \"\"\"Verify whether fn1(x) == fn2(x) for all x in inputs.\"\"\"\n",
    "    return all(fn1(x) == fn2(x) for x in inputs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Counting Barcodes\n",
    "\n",
    "> A typical barcode is pictured below. A valid barcode consists of alternating black and white stripes, where each stripe is either 1, 2, or 3 units wide. For a box that is *n* units wide, how many different valid barcodes are there?\n",
    "\n",
    "![barcode](https://help.shopify.com/assets/manual/sell-in-person/hardware/barcode-scanner/1d-barcode-4fbf513f48675746ba39d9ea5078f377e5e1bb9de2966336088af8394b893b78.png)\n",
    "\n",
    "We'll represent a unit as a character, `'B'` or `'W'`, and a barcode as a string of *n* units. The barcode above would start with `'BWWBW...'`.  A valid string is one without 4 of the same character/unit in a row; `valid_barcode` tests for this.\n",
    "\n",
    "We'll start with the  **enumerate and test** strategy: generate `all_strings` of *n* units and count how many are valid with `enumerate_barcodes`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def enumerate_barcodes(n) -> int: \n",
    "    \"\"\"Enumerate all barcodes of n units and count how many are valid.\"\"\"\n",
    "    return quantify(all_strings('BW', n), valid_barcode)\n",
    "\n",
    "def all_strings(alphabet, n): \n",
    "    \"\"\"All strings of length n over the given alphabet.\"\"\"\n",
    "    return {''.join(chars) for chars in product(alphabet, repeat=n)}\n",
    "\n",
    "def valid_barcode(code) -> bool: \n",
    "    \"\"\"A valid barcode does not have 4 or more of the same unit in a row.\"\"\"\n",
    "    return 'BBBB' not in code and 'WWWW' not in code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are all the strings of length 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'BBB', 'BBW', 'BWB', 'BWW', 'WBB', 'WBW', 'WWB', 'WWW'}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all_strings('BW', 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a table of counts of valid barcodes for small values of *n*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 1,\n",
       " 1: 2,\n",
       " 2: 4,\n",
       " 3: 8,\n",
       " 4: 14,\n",
       " 5: 26,\n",
       " 6: 48,\n",
       " 7: 88,\n",
       " 8: 162,\n",
       " 9: 298,\n",
       " 10: 548,\n",
       " 11: 1008,\n",
       " 12: 1854}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{n: enumerate_barcodes(n) for n in range(13)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This approach enumerates a lot of strings that can't possibly be valid. For example, there are 1024 strings of length 14 that start with `'BBBB...'` and none of them are valid. We could save a lot of time if we stopped generating such strings after we see the `'BBBB'`. \n",
    "\n",
    "The **incremental enumeration** strategy starts with all the valid strings of length 0 (there is only one, the empty string), and then repeats *n* times the process of appending one unit (`'B'` or `'W'`) to each string, if that append would be valid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def incremental_count_barcodes(n): \n",
    "    \"\"\"Count how many barcodes of length n are valid.\"\"\"\n",
    "    barcodes = {''}\n",
    "    for _ in range(n):\n",
    "        barcodes = extend_barcodes(barcodes)\n",
    "    return len(barcodes)\n",
    "\n",
    "def extend_barcodes(barcodes) -> set:\n",
    "    \"\"\"All valid ways to add one unit to each of a set of barcodes.\"\"\"\n",
    "    return {barcode + unit for barcode in barcodes for unit in 'BW'\n",
    "            if not barcode.endswith(3 * unit)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The four lines of code in the body of `incremental_count_barcodes` are exactly the pattern encapsulated in the `iterate` higher-order function. So we can rewrite `incremental_count_barcodes` with a one-line body:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def incremental_count_barcodes(n): \n",
    "    \"\"\"Count how many barcodes of length n are valid.\"\"\"\n",
    "    return len(iterate(extend_barcodes, {''}, n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Try it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1854"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "incremental_count_barcodes(12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify that the results are the same for small *n*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "same(enumerate_barcodes, incremental_count_barcodes, range(13))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's how I think about a more efficient approach:\n",
    "* I can use **abstract incremental enumeration**: find a representation that summarizes all the barcodes of length zero, then incremently extend that summary to get a summary of the barcodes of length 1. Do that *n* times.\n",
    "* At each step, the key information that needs to be in the summary is how many barcode units of the same color are at the *end* of a barcode: if it is 1 or 2, then we can add another instance of the same color to the end. If it is 3, we cannot. We can always add the opposite color, and then the resulting barcode will end in just one unit of the same color. \n",
    "* Thus, the summary will be a list of four counts: `[e0, e1, e2, e3]`, where `ei` gives the number of strings that end in `i` units of the same color. \n",
    "* To take this summary and extend it by one unit to make the next summary:\n",
    "   - For all the counts except `e3` we could add a unit of the same color; that would show up in the next higher position (e.g. a count of 4 in `e1` would show up as a count of 4 in `e2` in the next summary).\n",
    "   - For all the counts, we could add a unit of the opposite color; the sum of them would show up in `e1` of the next summary.\n",
    "* The function `abstract_barcodes(n)` does this update `n` times (using `iterate`) and in the end takes the sum of the four counts in the summary.\n",
    "* With **brute force enumeration**, incrementing *n* by 1 doubles the amount of work (because you double the number of candidate strings). With **incremental enumeration** there are fewer strings, but still an exponential number of them.\n",
    "* With the **abstract incremental** approach, incrementing *n* by 1 does a constant amount of work. Thus the total complexity is *O*(*n*), instead of *O*(2<sup>*n*</sup>) for **brute force enumeration** and (we will see later) approximately *O*(1.8<sup>*n*</sup>) for **incremental enumeration**.\n",
    "\n",
    "We can code that up as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "barcodes0 = [1, 0, 0, 0] # Summary of ending-counts of barcodes of length n=0\n",
    "\n",
    "def abstract_barcodes(n) -> int: \n",
    "    \"\"\"Count how many barcodes of length n are valid.\"\"\"\n",
    "    return sum(iterate(extend_barcodesummary, barcodes0, n))\n",
    "\n",
    "def extend_barcodesummary(barcodes) -> list:\n",
    "    \"\"\"Given a summary of barcodes of length n, build a summary for length n+1.\"\"\"\n",
    "    e0, e1, e2, e3 = barcodes\n",
    "    return [0, sum(barcodes) + e0, e1, e2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify that we get the same results for small values of *n*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "same(enumerate_barcodes, abstract_barcodes, range(20))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can examine the first few summaries:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: [1, 0, 0, 0],\n",
       " 1: [0, 2, 0, 0],\n",
       " 2: [0, 2, 2, 0],\n",
       " 3: [0, 4, 2, 2],\n",
       " 4: [0, 8, 4, 2],\n",
       " 5: [0, 14, 8, 4],\n",
       " 6: [0, 26, 14, 8],\n",
       " 7: [0, 48, 26, 14],\n",
       " 8: [0, 88, 48, 26],\n",
       " 9: [0, 162, 88, 48]}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{n: iterate(extend_barcodesummary, barcodes0, n) for n in range(10)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`abstract_barcodes` can quickly compute very big numbers that `enumerate_barcodes` could never handle:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 11.3 ms, sys: 643 µs, total: 12 ms\n",
      "Wall time: 12.6 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "3861431277625007961956955484353530119634001892816040917233932945064320273497370215771811960744678098449553175356862760450029708838175822242262792740296878964074883622671541479265048463512360941352413049264274485743297728357502930085506419846119753743423275462450713981257123756695327534235952507322555045959925039572403245743061549541274972562526816439217931608999532601457740681763142591939324781110768274782850152125981385364637513839024687081770052346957401052189529936883629883775724870785833452510126377097128195647948735625551805771200981065390268401761158588370204299868440790417559363818139283430755197453196664541025472104658523804014518931760254828135638415413408780736125999685589725526874318196976263624936793335541955083569139572617638693840543637407782446933562063756941909207810703824222697116352937601482868529114899390708691493432262019965964035273813939182009029538539757438413668036430833988535147478776959903569999055599703304516221455076523484352182404159659661240973630499557250950477386781288167303525408434907471599633608826344342446869378392685927230076723348547049789748491137890119623879128231595262082532286126660730784998797003448161418183918024344043613986269574364002761796194720086663633818066518383357158900007727428966795190669943187944515210398817044521469661682433819382541570464313514353180507280999817655346753846288267575488232309682752190042235927387740555053797529717247933281707578234957297940957985028202675009617273305705968728942732545640071819447202821044438874136038990729299397246489481363094787623333269036228204295737689912072742922999093173704662567170601571800655678495878408411165386708197339087266092115780445248022350264528021426918011958029075557108406192634108388793426193565093359228830473466263938925653882623387099940055081548579900911968982480432787324594136156465497071249090902373689488770079828664684036332439902542305885855063306802357476735193730805776865978477463443216064890685565824039494431583860044254937057151591772329166180799218273949130368000609439119706131185925513231524378443124711002954768565310478734594199963723940004083846148186188631535346393565417946059827530465095391420721324513264443555308798011729355327067365885381721113676766083967129134309202588682281325857855165574371421168078423586430302604691681703448297310856061359072019641696782664700526740970642410648738647199882852824774480069658314840218244137906558931392021855430239022442215072063184370394666755160595637651648014842470092745788026900633378764049390511584307920491613532339464196985013369306439276245475602674051266814071448421271626238787893306176669816773478303604652043427279626683524358370"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time abstract_barcodes(10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a big number! And it only took about 10 milliseconds! How many digits is the result?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2647"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def digits(n: int) -> int: return len(str(n))\n",
    "\n",
    "digits(abstract_barcodes(10000))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the number of valid barcodes of length 10,000 is more than 10<sup>2646</sup>. \n",
    "\n",
    "We can apply the **shoulders of giants** strategy and search for the first few numbers in the series, [\"1, 2, 4, 8, 14, 26, 48\"](https://www.google.com/search?q=%221%2C+2%2C+4%2C+8%2C+14%2C+26+48%22), and get as the first result the page [oeis.org/A135491](https://oeis.org/A135491), titled *Number of ways to toss a coin n times and not get a run of four*, which sounds right. [OEIS](https://oeis.org/), the On-Line Encyclopedia of Integer Sequences®, is a fantastic resource for all those who count things, and often shows up in searches like this."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "# Problem: Counting Student Records (Late/Absent/Present)\n",
    "\n",
    "> Students at a school must check in with the guidance counselor if they have three total absences, or three consecutive late days. Each student's attendance record consists of a string of 'A' for absent, 'L' for late, or 'P' for present. For example: \"LAPPLAA\" requires a meeting (because there are three absences), and \"LAPPLAL\" is OK (there are three late days, but they are not consecutive). How many attendance records of length *n* days are OK?\n",
    "\n",
    "The **brute force enumeration** strategy applies in a similar way to the previous problem. Define what it means for a record to be `ok`, generate all the strings of length *n*, and count how many of them are `ok`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ok(record: str) -> bool: \n",
    "    \"\"\"Is this student record OK? (Not 3 absences, nor 3 consecutive lates.)\"\"\"\n",
    "    return record.count('A') < 3 and 'LLL' not in record\n",
    "\n",
    "def enumerate_count_ok(n) -> int:\n",
    "    \"\"\"How many attendance records of length n are ok?\"\"\"\n",
    "    return quantify(all_strings('LAP', n), ok)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 1,\n",
       " 1: 3,\n",
       " 2: 9,\n",
       " 3: 25,\n",
       " 4: 67,\n",
       " 5: 171,\n",
       " 6: 419,\n",
       " 7: 994,\n",
       " 8: 2296,\n",
       " 9: 5188,\n",
       " 10: 11510}"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{n: enumerate_count_ok(n) for n in range(11)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks good, but there are 3<sup>*n*</sup> strings of length *n*, so for large values of *n*\n",
    "we will need a more efficient strategy.\n",
    "\n",
    "* The **abstract incremental enumeration** strategy is again applicable: find a representation that summarizes all the attendance records of length zero, then incremently extend the length by 1, and do that *n* times. The function `abstract_count_ok` implements this approach, again using `iterate`.\n",
    "* What is in the summary? A list of all `ok` records is too much. A simple count of the number of `ok` records is not enough. Instead, we need several different counts, for several different classes of records. Each class is defined by the number of `'A'` characters in the records, and the number of consecutive `'L'` characters at the *end* of the records, because these are the two things that determine whether the string will be `ok` or not `ok` when we add letters to the end).  So the summary can be represented as  a `Counter` of the form `{(A, L): count, ...}`. For example the summary `{(1, 2): 3, ...}` means that there are 3 `ok` records that contain one `'A'` and end in two `'L'`s. They records aren't explicitly named in the summary (that's why the summary can be efficient), but they would be `{'APLL', 'LALL', 'PALL'}`.\n",
    "* For *n* = 0, the summary is `{(0, 0): 1}`: one record of length 0, the empty string, which has no `'A'` in it and no `'L'` at the end. \n",
    "* The function `extend_one_day` says that we can add an `'A'` to any string that doesn't already have two `'A'`s; we can add an `L` to any string that doesn't already end in 2 `'L'`s; and we can add a `'P'` to any string."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "records0 = Counter({(0, 0): 1})\n",
    "\n",
    "def abstract_count_ok(n) -> int:\n",
    "    \"\"\"How many attendance records of length n are ok?\"\"\"\n",
    "    return total(iterate(extend_records, records0, n))\n",
    "\n",
    "def extend_records(records: Counter) -> Counter:\n",
    "    \"\"\"Given a summary of records of length n in the form {(A, L): count}, \n",
    "    build a summary of records of length n + 1.\"\"\"\n",
    "    next_records = Counter()\n",
    "    for (A, L), count in records.items():\n",
    "        if A < 2: next_records[A + 1, 0] += count # add an 'A'\n",
    "        if L < 2: next_records[A, L + 1] += count # add an 'L'\n",
    "        next_records[A, 0] += count               # add a  'P'\n",
    "    return next_records"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify that `abstract_count_ok` gets the same counts as `enumerate_count_ok`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "same(abstract_count_ok, enumerate_count_ok, range(11))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the first few summaries of records of length *n*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: Counter({(0, 0): 1}),\n",
       " 1: Counter({(1, 0): 1, (0, 1): 1, (0, 0): 1}),\n",
       " 2: Counter({(2, 0): 1,\n",
       "          (1, 1): 1,\n",
       "          (1, 0): 3,\n",
       "          (0, 2): 1,\n",
       "          (0, 0): 2,\n",
       "          (0, 1): 1}),\n",
       " 3: Counter({(2, 1): 1,\n",
       "          (2, 0): 5,\n",
       "          (1, 2): 1,\n",
       "          (1, 0): 8,\n",
       "          (1, 1): 3,\n",
       "          (0, 0): 4,\n",
       "          (0, 1): 2,\n",
       "          (0, 2): 1})}"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{n: iterate(extend_records, records0, n) for n in range(4)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, the entry for `1:` says there are 3 total  strings of length one; one of which contains 1 `A` and does not end in `L`; one of which does not contain an `A` and does end in one `L`, and one of which has neither `A` nor `L`. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course `abstract_count_ok` can go *way* beyond what we could do with `enumerate_count_ok`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "67915092244946245526761121539961065667354710339484389403398502322665903317227269563431987985548850569926132622865368781540197647175409335204446318662764534903563777210453736211255159314858993923969786046819296950933246810634303897405512752272032426651642064888476180415506912693513909342933046331038474619509081763784765550352067334165901573244296556383159625556197148091705165182887658871588760415124598528254895046939766294908991602550833585558872584109925815975563950518270492886424386418887038225367557977580598940456529146317404617930359322164816128982301399847355494961116547323299531092346924090578848490063810929726294235553624714771568277381300778770044740403839922522259903425992438105556328232190837523076945505837289570218129815218657504553108186081068344280320487112129980538558330653940927620274198611870978884765113302222147574340037452775946171158479481569109577479495284213988633469847097599897091262353972939940610742973967768665639608260138071788253672077302950170544573659098025328249386699113294464533042986243917770015922628745694209690218902434289336799469726583915592005603572447725365675690796428672725349270535435510404530717474913302992334196693282513211071037040630045078690538568866750760269179165191419371245126202447626783645521606741196697550515070422233997330690541032248272343548507506077080632447162616725116827906765346068097284144021799385759481910334161557352438965487177479943712008492652355168983678944082674396835412877102701320156716226396792442810305911973460027845758272368905315276808267770968165158601228698100671353970870820495437717365750878332077859999742716730188128437024395707454371180819695507850153005101353643954347256441396183843055876562258195012627902932249519909390770919657440052840040197024105521065595910086035192717059260479461379754157116341147835641061318945374959021427192475455471423853727585174268760196347849181893783510713204119256033951950166887204536135896283786882669204124059336236100878907356839002912187712588976111620701336082598908391856751671391768146332236193267088262535153912806255587921455005824532343009909950356632076138892829754315844530603374067069772173161102995334001987128851407630096692078344942645715716941016099638170178149988009446531234694691215930910342787376545269610229494026686126796325578380820170813105262828560980051029484551420232365099464095174864505651859406571715076351131405366919080277275366319781500406689243537529390550548233759064314753797975451923281566305080203709436529025198215417739606729647754826931823619915850794244671776741088055751566519186191628709222778000007816869534445057653832155401753050275129786525546204005123078586341515867"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abstract_count_ok(10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That looks like a roughly similar number of digits as `abstract_barcodes(10000)`; let's compare:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2654, 2647)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 10000\n",
    "digits(abstract_count_ok(N)), digits(abstract_barcodes(N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So `abstract_count_ok(10000)` is only 7 digits more than `abstract_barcodes(10000)`, out of 2654. \n",
    "\n",
    "I'm curious how close the two series are in general. I'll plot them on a log-plot and compare with $2^n$ and $1.8^n$ (I'll only go up to $n = 366$ days to avoid overflow, but the straight lines look the same at any $n$ values):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def logplots(X, fns, xlabel='', ylabel=''): \n",
    "    \"\"\"Given fns={label: fn,...} plot fn(x) vs x for each fn.\"\"\"\n",
    "    plt.yscale('log'); plt.xlabel(xlabel); plt.ylabel(ylabel)\n",
    "    for label in fns:\n",
    "        plt.plot(X, [fns[label](x) for x in X], '-',  label=label)\n",
    "    plt.legend()\n",
    "\n",
    "logplots(range(1, 367, 5), {\n",
    "    '2^n': (2).__pow__,\n",
    "    'abstract_count_ok': abstract_count_ok,\n",
    "    'abstract_barcodes': abstract_barcodes,\n",
    "    '1.8^n': 1.8.__pow__}, \n",
    "    'Number of days', 'Number of Records')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Pirates Counting Coconuts\n",
    "\n",
    "The 538 Riddler poses [this problem](https://fivethirtyeight.com/features/pirates-monkeys-and-coconuts-oh-my) (slightly edited):\n",
    "\n",
    ">Seven pirates wash ashore on a deserted island. They gather coconuts into a central pile. As the sun sets, they all go to sleep.\n",
    ">\n",
    ">One pirate wakes up in the middle of the night. Being greedy, this pirate decides to take some coconuts from the pile and hide them for himself. As he approaches the pile, though, he notices a monkey watching him. To keep the monkey quiet, the pirate tosses it one coconut from the pile. He then divides the rest of the pile into seven equal bunches and hides one of the bunches in the bushes. Finally, he recombines the remaining coconuts into a single pile and goes back to sleep.\n",
    ">\n",
    ">In succession, each of the pirates does the same routine. In the morning, the pirates split the pile into seven equal bunches and take one bunch each. (The monkey does not get one this time.)\n",
    ">\n",
    ">If there were *c* coconuts in the pile originally, what is the smallest possible value of *c*? What is *c* if there are *p* pirates?\n",
    "\n",
    "At least for *p* = 7, it seems like an **enumerate and test** approach should work, where we enumerate the number of coconuts, *c* and check if each amount can be evenly divided multiple times as the story dictates. The function `coconuts_divisible` does the check and `enumerate_coconuts` enumerates values of *c* and returns the first one that works. We can be somewhat clever in the enumeration: we know that the first pirate throws one coconut to the monkey and then the pile is divisible by *p*. So the only numbers we need to consider for *c* are of the form *kp* + 1, which we write in Python as `range(1, maxsize, p)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def enumerate_coconuts(p=7) -> int:\n",
    "    \"\"\"Find the smallest number of coconuts, c, that makes the story work for p pirates.\"\"\"\n",
    "    return next(c for c in range(1, maxsize, p) if coconuts_divisible(p, c))\n",
    "\n",
    "def coconuts_divisible(p, c) -> bool:\n",
    "    \"\"\"Can p pirates divide c coconuts evenly, following the steps of the story?\"\"\"\n",
    "    for pirate in range(p):   # Each successive pirate\n",
    "        c -= 1                # tosses the monkey one coconut\n",
    "        if not divides(p, c): # divides the rest of the pile into p equal bunches\n",
    "            return False\n",
    "        c -= c // p           # and hides one bunch\n",
    "    return divides(p, c)      # In the morning they split the pile evenly.\n",
    "        \n",
    "def divides(P, C) -> bool: return C % P == 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "823537"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "enumerate_coconuts(7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a big pile of coconuts. Let's see how many coconuts are needed with fewer pirates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 3, 25, 765, 3121, 233275, 823537]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[enumerate_coconuts(p) for p in range(1, 8)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **shoulders of giants** search \n",
    "[\"1, 3, 25, 765, 3121\"](https://www.google.com/search?q=%221+3+25+765+3121%22&oq=%221+3+25+765+3121%22)\n",
    "yields as first result [oeis.org/A002021](https://oeis.org/A002021), titled \"Pile of coconuts problem\" so that is definitely the right page. The page gives a **formula calculation**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_coconuts(p) -> int:\n",
    "    \"\"\"Find the smallest number of coconuts that makes the story work for p pirates.\"\"\"\n",
    "    return (p - 1) * (p ** p - 1) if divides(2, p) else p ** p - p + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can verify the formula and use it to explore bigger numbers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 1,\n",
       " 2: 3,\n",
       " 3: 25,\n",
       " 4: 765,\n",
       " 5: 3121,\n",
       " 6: 233275,\n",
       " 7: 823537,\n",
       " 8: 117440505,\n",
       " 9: 387420481,\n",
       " 10: 89999999991,\n",
       " 20: 1992294399999999999999999981,\n",
       " 30: 5970842830744820999999999999999999999999999971,\n",
       " 40: 471481069649705378135408639999999999999999999999999999999999999961,\n",
       " 50: 435207425653061363846063613891601562499999999999999999999999999999999999999999999999951,\n",
       " 60: 2883547000860666191870042384152701628073990160383999999999999999999999999999999999999999999999999999999999941}"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "assert same(calculate_coconuts, enumerate_coconuts, range(1, 8))\n",
    "\n",
    "{p: calculate_coconuts(p) for p in [*range(1, 10), *range(10, 61, 10)]}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "998999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999001"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calculate_coconuts(1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a crazy number. Almost $10^{3003}$, and you surely noticed all the \"9\"s and the \"001\" at the end, but did you notice the lone \"8\" in the third digit? What's that doing there? Actually it's simple: you can think of the formula as first giving us $999 \\times 1000^{1000}$, which is \"999\" followed by 3000 \"0\"s, and then subtracting 999, so that third \"9\" is where we borrow 1, and leads to all the following \"0\"s becoming \"9\"s."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Counter({'9': 2999, '8': 1, '0': 2, '1': 1}), 1.0)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Counter(str(calculate_coconuts(1000))), log(10, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extension: Non-Minimal Numbers of Coconuts\n",
    "\n",
    "I'm curious what *other* numbers of coconuts *c* are possible, not just the minimal number for each  *p*. Below I compute the 7 smallest values of *c* for each number of pirates *p* in the range 1-7:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: [1, 2, 3, 4, 5, 6, 7],\n",
       " 2: [3, 11, 19, 27, 35, 43, 51],\n",
       " 3: [25, 106, 187, 268, 349, 430, 511],\n",
       " 4: [765, 1789, 2813, 3837, 4861, 5885, 6909],\n",
       " 5: [3121, 18746, 34371, 49996, 65621, 81246, 96871],\n",
       " 6: [233275, 513211, 793147, 1073083, 1353019, 1632955, 1912891],\n",
       " 7: [823537, 6588338, 12353139, 18117940, 23882741, 29647542, 35412343]}"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def kcoconuts(p=7, k=7) -> [int]:\n",
    "    \"\"\"Find the k smallest number of coconuts, c, that makes the story work for p pirates.\"\"\"\n",
    "    return list(islice((c for c in range(1, maxsize, p) if coconuts_divisible(p, c)), k))\n",
    "\n",
    "{p: kcoconuts(p) for p in range(1, 8)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These numbers look like they're in arithmetic sequences (the same difference between each pair of adjacent numbers). I can test that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def arithmetic_sequence(numbers) -> tuple:\n",
    "    \"\"\"Are the numbers in an arithmetic sequence? Return the first and the differences.\"\"\"\n",
    "    deltas = {numbers[i] - numbers[i - 1] for i in range(1, len(numbers))}\n",
    "    return (numbers[0], deltas)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: (1, {1}),\n",
       " 2: (3, {8}),\n",
       " 3: (25, {81}),\n",
       " 4: (765, {1024}),\n",
       " 5: (3121, {15625}),\n",
       " 6: (233275, {279936}),\n",
       " 7: (823537, {5764801})}"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{p: arithmetic_sequence(kcoconuts(p)) for p in range(1, 8)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That confirms it (at least up to 7), and I see a pattern in the deltas: we have 2: 8 = 2<sup>3</sup>, 3: 81 = 3<sup>4</sup>, 4: 1024 = 4<sup>5</sup>, etc. I can verify the pattern (at least up to 7):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 1, 2: 8, 3: 81, 4: 1024, 5: 15625, 6: 279936, 7: 5764801}"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{p: p ** (p + 1) for p in range(1, 8)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we can say the complete set of valid number of coconuts for a given number of pirates *p* is:\n",
    "    \n",
    "- $(p - 1) \\times (p^p - 1) + k \\times p^{p+1}$ when $p$ is even, for any nonnegative integer $k$\n",
    "- $p^p - p + 1  + k \\times p^{p+1}$ when $p$ is odd, for any nonnegative integer $k$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extension: Who Gets What?\n",
    "\n",
    "We can also count who gets what shares of the coconuts, by annotating the story. The monkey is individual 0, and the pirates are 1 to *p*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "def coconuts_shares(p) -> dict:\n",
    "    \"\"\"Assuming p pirates divide coconuts evenly according to the story, who gets what share?\"\"\"\n",
    "    c       = calculate_coconuts(p)\n",
    "    monkey  = 0\n",
    "    pirates = range(1, p + 1)\n",
    "    shares  = Counter()\n",
    "    def to(who, what): shares[who] += what; return what\n",
    "    for pirate in pirates:       # Each successive pirate\n",
    "        c -= to(monkey, 1)       # tosses the monkey one coconut\n",
    "        assert divides(p, c)     # divides the rest of the pile into p equal bunches\n",
    "        c -= to(pirate, c // p)  # and hides one bunch     \n",
    "    for pirate in pirates:       # In the morning they split the pile evenly.\n",
    "        to(pirate, c // p)\n",
    "    return dict(shares)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 7,\n",
       " 1: 157638,\n",
       " 2: 140831,\n",
       " 3: 126425,\n",
       " 4: 114077,\n",
       " 5: 103493,\n",
       " 6: 94421,\n",
       " 7: 86645}"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coconuts_shares(7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert total(_) == enumerate_coconuts(7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is best to be the first pirate; every other pirate does successively worse, with the last one getting about half of the first's share. The monkey always gets *p* coconuts from *p* pirates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: {0: 1, 1: 0},\n",
       " 2: {0: 2, 1: 1, 2: 0},\n",
       " 3: {0: 3, 1: 10, 2: 7, 3: 5},\n",
       " 4: {0: 4, 1: 251, 2: 203, 3: 167, 4: 140},\n",
       " 5: {0: 5, 1: 828, 2: 703, 3: 603, 4: 523, 5: 459},\n",
       " 6: {0: 6, 1: 51899, 2: 45419, 3: 40019, 4: 35519, 5: 31769, 6: 28644}}"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{p: coconuts_shares(p) for p in range(1, 7)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Counting Rhyme Schemes\n",
    "\n",
    "Here's another problem:\n",
    "\n",
    "> How many rhyme schemes are there for a *k* line poem? 'ABAB' is a valid rhyme scheme, as is 'ABBA', but 'BAAB' is not: in a valid rhyme scheme the first occurrences of each letter forms a prefix of the alphabet.\n",
    "\n",
    "Let's first make sure we understand what counts as a valid rhyme scheme. Given a string, first task pick out the **first occurrences** of letters. For example, for the string `\"ABAB\"`, we read left-to-right, recording each time we see a letter for the first time; that gives us `\"AB\"`; the subsequent occurrences of `'A'` and `'B'` don't matter. Given `\"BAAB\"`, the first occurrences are `\"BA\"`. Now we have to decide if these first occurrences form a **prefix of the alphabet**,  `\"ABCD...\"`. For `\"AB\"` the answer is yes, but for `\"BA\"` the answer is no. \n",
    "\n",
    "Before we get to the counting, below are the basic concepts.  Since we will want to go beyond *k* = 26, I will by default make the alphabet be the non-negative integers, so \"letters\" are integers. Use `alphabet` if you want strings of actual letters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "alphabet = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\n",
    "\n",
    "def is_rhyme_scheme(s, alphabet=range(maxsize)) -> bool: \n",
    "    \"\"\"Do the first occurrences of letters in `s` form a prefix of the alphabet?\"\"\"\n",
    "    return is_prefix(first_occurrences(s), alphabet)\n",
    "\n",
    "def is_prefix(short, long) -> bool:\n",
    "    \"\"\"Is the first argument a prefix of the second?\"\"\"\n",
    "    return all(S == L for S, L in zip(short, long))\n",
    "\n",
    "def first_occurrences(s) -> tuple: \n",
    "    \"\"\"The first occurrences of each character, in the order they appear.\"\"\"\n",
    "    return tuple(Counter(s)) # Counters preserve order of elements"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_rhyme_scheme('ABAB', alphabet)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_rhyme_scheme('BAAB', alphabet)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_rhyme_scheme('ABBACABBADABBA', alphabet)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_rhyme_scheme('ABBACABBADABBADO', alphabet)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can count the number of valid strings by **brute force enumeration**: generate all possible strings of length $k$ and check each one with `is_rhyme_scheme`. The complexity of this algorithm is $O(k^{k+1})$, because there are $k^k$ strings, and to validate a string requires looking at $k$ characters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 1, 1: 1, 2: 2, 3: 5, 4: 15, 5: 52, 6: 203, 7: 877}"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def enumerate_rhyme_schemes(k) -> int: \n",
    "    \"\"\"Enumerate all possible strings and count the number of valid rhyme schemes.\"\"\"\n",
    "    strings = product(range(k), repeat=k)\n",
    "    return quantify(strings, is_rhyme_scheme)\n",
    "\n",
    "{k: enumerate_rhyme_schemes(k) for k in range(8)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's think about an **abstract incremental enumeration** strategy.\n",
    "As with previous problems, I need a *summary* of the relevant information for strings of length *&nbsp;k*, to help me calculate the relevant information for length *&nbsp;k*+1. I know that if I have a valid string of length *&nbsp;k* with *d* distinct characters in it, then I can extend it by one character in *d* ways by repeating any of those *d* characters, or I can introduce a first occurrence of character number *d+1* (but I can do that in just 1 way). I can't validly introduce any other character. So a good summary would be a Counter of `{d: count, ...}`. We start with strings of length 0, which have a summary of `{0: 1}` (one string, the empty string, with 0 distinct characters) and we get this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "def abstract_rhyme_schemes(k) -> int: \n",
    "    \"\"\"Count the number of strings that are valid rhyme schemes.\"\"\"\n",
    "    return total(iterate(extend_rhymes, Counter({0: 1}), k))\n",
    "\n",
    "def extend_rhymes(counts) -> Counter:\n",
    "    \"\"\"Given a summary of the form {d: count}, return summary for one character more.\"\"\"\n",
    "    return Counter({d: d * counts[d] + counts[d - 1] \n",
    "                    for d in range(len(counts) + 1)})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see the summary for strings of length *k* = 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({0: 0, 1: 1, 2: 3, 3: 1})"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "iterate(extend_rhymes, Counter({0: 1}), 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This says that for strings of length 3, there is 1 valid string with 1 distinct letter (which happens to be the string `'AAA'`, but the summary doesn't say that),  3 valid strings with 2 distinct letters (`'AAB', 'ABA', 'ABB'`) and 1 valid string with 3 distinct letters (`'ABC'`).\n",
    "\n",
    "We can show that this approach gives the same results as brute force enumeration (at least up to 7):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "same(enumerate_rhyme_schemes, abstract_rhyme_schemes, range(8))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can see that this approach can handle much bigger values of *k*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "29899013356824084214804223538976464839473928098212305047832737888945413625123259596641165872540391578300639147082986964028021802248993382881013411276574829121155811755170830666039838837273971971676782389800810361809319250755399325279656765435255999301529770267107281619733800281695881540007577899106878679451165492535930459233713316342551545242815802367257284852612201081016386308535990145447341800455472334713864080523978960296365736999295932080550928561633025800627524911700149562106895897725047744775812241800937310491797818107578233924187312824632629095993832334781713007323483688294825326897450386817327410532925074613888321264138083842196202242956001314953449497244271843922741908252107652201346933889741070435350690242062001522697855278356012055718392851567813397125419144780476479197990921602015873703820769182603836788465785093563686025690269802153802436873530877006737154523895273029510238745997356292232631282773748762989386003970214423843947094021177989737557020369751561595003372955621411858485959813344799967960196238368337022346946771703060269288691694028444791203978533454759410587065022546491518871238421560825907135885619221776405898771057270555581449229994215739476758785884545723062263992367750091319644861547658472282284005892044371587560711880627741139497818835632120761570174928529697397267899554407350161283097123211048049269727655279783900702416095132827766428865017653366696304131436690232979453876337599721772897049270230544262611264917393374756384152784943607952408782612639220380791445272655004475989064276373713608901650681165467490310898804916827069427310961109285035545084791339423266482359955663377201515204340817580915468489969181643341007197836481461051798995640789292580146918580703759556634019451731530034209189203377522668309771129566108101617727442045637098112678864654309987785463307376544339506878267267349348171320834971956806668304099159992067385998690820326902473886782781499414773179"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abstract_rhyme_schemes(1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a big number. \n",
    "\n",
    "A search for [\"1, 1, 2, 5, 15\"](https://www.google.com/search?q=%221%2C+1%2C+2%2C+5%2C+15%22) shows that this sequence is called [Bell numbers](https://en.wikipedia.org/wiki/Bell_number), and that they have lots of applications.\n",
    "\n",
    "## Version 2: Recursion\n",
    "\n",
    "Another way to keep track of the number of valid strings of length *&nbsp;k* with *&nbsp;d* distinct characters is with a recursive counting function, which I will call `count_rhymes(k, d)`. There are three cases: \n",
    "\n",
    "- `count_rhymes(0, 0)` is 1, because the empty string is valid (and contains no distinct characters). \n",
    "- `count_rhymes(k, d)` is 0 when `k` is negative (because there is no such thing as a negative length string) or less than `d` (because a string can't contain more distinct characters than its length). \n",
    "- Otherwise, there are `d` ways to add an existing letter to each of the strings of length `k - 1`, and there is one way to add a new letter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "@lru_cache(None)\n",
    "def count_rhymes(k, d) -> int:\n",
    "    \"\"\"Count the number of valid strings of length k, that use d distinct characters.\"\"\"\n",
    "    return (1 if (k == 0 == d)    else\n",
    "            0 if (k < 0 or k < d) else\n",
    "            d * count_rhymes(k - 1, d) + count_rhymes(k - 1, d - 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that I used `lru_cache`, to avoid having to re-compute intermediate results.\n",
    "\n",
    "Now I can define `calculate_rhyme_schemes(k)` as the sum of `count_rhymes(k, d)` over all values of `d`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_rhyme_schemes(k) -> int: \n",
    "    \"\"\"Count the number of strings that are valid rhyme schemes.\"\"\"\n",
    "    return sum(count_rhymes(k, d) for d in range(k + 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's validate this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "same(abstract_rhyme_schemes, calculate_rhyme_schemes, range(101))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Counting Rectangle Sets\n",
    "\n",
    "This problem is covered in depth in [another notebook](Golomb-puzzle.ipynb), so here I present just the part that has to do with counting things:\n",
    "\n",
    "> Say you’re given the following challenge: create a set of five rectangles that have sides of length 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 units. You can combine sides in a variety of ways: for example, you could create a set of rectangles with dimensions 1 x 3, 2 x 4, 5 x 7, 6 x 8 and 9 x 10. How many different sets of five rectangles are possible?\n",
    "\n",
    "This is a basic [combinatorics](http://en.wikipedia.org/wiki/Combinatorics) problem. I will present *three* methods to calculate the number of sets. If all goes well they will give the same answer. The example set of rectangles given in the problem was\n",
    "\n",
    "     {1 × 3, 2 × 4, 5 × 7, 6 × 8, 9 × 10}\n",
    "    \n",
    "and in general it would be\n",
    "\n",
    "     {A × B, C × D, E × F, G × H, I × J}\n",
    "\n",
    "The question is: how many distinct ways can we assign the integers 1 through 10 to the variables A through J?\n",
    "    \n",
    "**Method 1: Count all permutations and divide by repetitions:** There are 10 variables to be filled, so there are 10! = 3,628,800 permutations.  But if we fill the first two variables with 1 × 3, that is the same rectangle as 3 × 1. So divide 10! by 2<sup>5</sup> to account for the fact that each of 5 rectangles can appear 2 ways.  Similarly, if we fill A and B with 1 × 3, that yields the same set as if we filled C and D with 1 × 3.  So divide again by 5! (the number of permutations of 5 things) to account for this.\n",
    "That gives us:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "945.0"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(10) / 2 ** 5 / factorial(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(It is always a relief when this \"count and divide\" method comes out to a whole number.)\n",
    "\n",
    "**Method 2: Count without repetitions**: in each rectangle of the example set the smaller component is listed first, and in each set, the rectangles with smaller first components are listed first. Without loss of generality, let's assume that all rectangle sets must be of this form, and count how many sets there are that respect this ordering.  We'll work from left to right.  How many choices are there for variable A?  *Only one!* A must always be 1, because we agreed that the smallest number comes first. Then, given A, there are 9 remaining choices for B.  For C, given A and B, there is again only one choice: C must be the smallest of the remaining 8 numbers.  That leaves 7 choices for D, 5 for F, 3 for H and 1 for J. So:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "945"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "9 * 7 * 5 * 3 * 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(It is always a relief when two methods give the same answer.)\n",
    "          \n",
    "**Method 3: Write a program to enumerate and test:** We'll generate all permutations of sides, and then check which ones are valid rectangle sets: they must have the first element of each pair less than the second (i.e. A < B, C < D, ...) and the pairs must be sorted, which is equivalent to saying their first elements are sorted (i.e. A < C < E < G < I)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "945"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def valid_rectangle_set(sides) -> bool:\n",
    "    \"\"\"Are the sides ordered according to convention?\"\"\"\n",
    "    A,B, C,D, E,F, G,H, I,J = sides\n",
    "    return A < B and C < D and E < F and G < H and I < J and A < C < E < G < I\n",
    "\n",
    "quantify(permutations(range(1, 11)), valid_rectangle_set)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(It is a relief that once again we get the same answer.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Counting Paths on a Grid\n",
    "\n",
    "> In a grid, how many paths are there from the upper left to the lower right corner, making only rightward or downward moves?\n",
    "     \n",
    "Here is an example 11 × 6 grid, with three of the possible paths:\n",
    "\n",
    "     ----------+     |..........     --+........\n",
    "     ..........|     |..........     ..+-+......\n",
    "     ..........|     +--+.......     ....+-+....\n",
    "     ..........|     ...|.......     ......+-+..\n",
    "     ..........|     ...|.......     ........|..\n",
    "     ..........|     ...+-------     ........+--\n",
    "     \n",
    "We can use the same three methods as the previous problem:\n",
    "\n",
    "**Method 1: Count all permutations and divide by repetitions:** Any path on this grid must consist of 10 right and 5 down moves, but they can appear in any order. Arranging 15 things in any order gives 15! = 1,307,674,368,000 possible paths. But that counts all the moves as being distinct, when actually the 10 right moves are indistinguishable, as are the 5 down moves, so we need to divide by the number of ways that they can be arranged. That gives us:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3003"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(15) // (factorial(10) * factorial(5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Method 2: Count without repetitions**: Another way to look at it is that there will be 15 total moves, so start with all 15 being \"right\" moves and  then choose 5 of them to become \"down\" moves. So the answer is (15 choose 5), which happens to lead to the same formula we just used:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3003"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def choose(n, k) -> int: return factorial(n) // (factorial(n - k) * factorial(k))\n",
    "\n",
    "choose(15, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Method 3: Write a program to count the paths:** The function `count_paths(start, goal)` counts the number of paths from the start location to the goal location, where a location is an `(x, y)` pair of integers.\n",
    "In general, the number of paths to the goal is the number of paths to the location just to the left of the goal, plus the number of paths to the location just above the goal. But there are two special cases: there is only one path (the empty path) when the start is equal to the goal, and there are zero paths when the goal is an invalid destination (one with a negative coordinate)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3003"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@lru_cache(None)\n",
    "def count_paths(start=(0, 0), goal=(5, 10)) -> int:\n",
    "    \"\"\"Number of paths to goal, using only 'right' and 'down' moves.\"\"\"\n",
    "    (x, y) = goal\n",
    "    return (1 if goal == start else\n",
    "            0 if x < 0 or y < 0 else\n",
    "            count_paths(start, (x - 1, y)) + \n",
    "            count_paths(start, (x, y - 1)))\n",
    "    \n",
    "count_paths()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Even though `count_paths` is slower than the `choose` calculation, it can still handle reasonably large grids:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90548514656103281165404177077484163874504589675413336841320"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 100\n",
    "assert count_paths(goal=(N, N)) == choose(2 * N, N)\n",
    "count_paths(goal=(N, N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why bother with the recursive function `count_paths` when the `choose` formula works so well? Good question. One reason is **checking**: the two different approaches validate each other by giving the same answer. Another reason is that we can modify `count_paths` to handle things that are hard to do with just the formula. For example, what if we have a grid with some obstacles in it?  I'll define a `Grid` constructor, which adopts the convention that the input is a string of rows, where a `'.'` character within a row is a passable square, and all other (non-whitespace) characters are impassible barriers. Then `count_grid_paths` finds the number of paths on a grid from start to goal (by default, from upper left to lower right):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Grid(text): return tuple(text.split())\n",
    "\n",
    "passable = '.'\n",
    "\n",
    "@lru_cache(None)\n",
    "def count_grid_paths(grid, start=(0, 0), goal=None) -> int:\n",
    "    \"\"\"Number of paths from start to goal on grid, using only 'right' and 'down' moves.\"\"\"\n",
    "    if goal is None: goal = (len(grid[-1]) - 1, len(grid) - 1) # bottom right\n",
    "    (x, y) = goal\n",
    "    return (1 if goal == start else\n",
    "            0 if x < 0 or y < 0 or grid[y][x] != passable else\n",
    "            count_grid_paths(grid, start, (x - 1, y)) + \n",
    "            count_grid_paths(grid, start, (x, y - 1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can verify that we get the same answer on the 11 by 6 empty grid:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3003"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_grid_paths(Grid(\"\"\"\n",
    "...........\n",
    "...........\n",
    "...........\n",
    "...........\n",
    "...........\n",
    "...........\n",
    "\"\"\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a grid where there are only two paths around the walls:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_grid_paths(Grid(\"\"\"\n",
    "...........\n",
    ".........|.\n",
    ".........|.\n",
    ".........|.\n",
    ".--------+.\n",
    "...........\n",
    "\"\"\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we put a hole in the wall, there should be many paths (but less than 3003 because most of the wall remains):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "122"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_grid_paths(Grid(\"\"\"\n",
    "...........\n",
    ".........|.\n",
    ".........|.\n",
    ".........|.\n",
    ".------.-+.\n",
    "...........\n",
    "\"\"\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are a couple of bigger examples, courtesy of [ascii-art-generator.org](https://www.ascii-art-generator.org/):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "627084695807418"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_grid_paths(Grid(\"\"\"\n",
    "....................................................................................................\n",
    "...................................NK0OkdddolcccccccccclloddxkO0XN..................................\n",
    "............................X0kdlc:cccccclodxllxxxkkxdloddolccccccccodkKN...........................\n",
    ".......................N0xl:cccllcco0.XXXXXX.l,O.NNNNo,kXXXXXXXNklllolccccokKN......................\n",
    "..................X.0dc::loool:,XXXdXXXXXXXXN:.X''X''X.dXXXXXXX.KcXX';coooolcclkX...................\n",
    ".................Kd.:cooo:'X.....'O.XXXXXXXXX;.........oXXXXXXXXXNo......X,lddoc:ckX................\n",
    "...............Oc,c.oc'..........l.XXXXXXXXXOX.........:NXXXXXXXXX0'.........X;oxoc;oK..............\n",
    ".............0c,lxl..............,O.XXXXXXXK;..........XlNXXXXXXXXlX............X;dxc,oX............\n",
    "...........Nx':xlX................X;dOKXX0d'.............,d0XK0xl'.................,xx;;O...........\n",
    "...........dXlk,......................XXXX.................XXX......................Xl0:xO..........\n",
    "..........OX:0:.......................................................................d0x;X.........\n",
    "..........lXdOX.......................................................................;XlXk.........\n",
    "..........lXdOX.......................................................................;XlXk.........\n",
    ".........XkX:0:...............XX....................................XX................dK,;K.........\n",
    "..........NoXlk,...........XoOKK0xc'.....X'..............XXX....X;okKXKO:X..........Xl0:Xk..........\n",
    "...........XoX:xcX.........o.XXXXXXNk:X,kNNO:..........Xl0NKoX'oK.XXXXXXX:.........'xk;,k...........\n",
    "...........X.k;'lo;X.......l.XXXXXXXX.XXXXXXNxX.......,O.XXX.XNXXXXXXXXXK,.......'oxc'cK............\n",
    ".............XNx;,:lc,.....Xd.XXXXXXXXXXXXXXX.O'.....cXXXXXXXXXXXXXXXXXX:.....Xcoo:,cO..............\n",
    "...............XNOl;;:c:;X..XlXXXXXXXXXXXXXXXXX0,...oNXXXXXXXXXXXXXXX.O,..X;c.lc;:dK................\n",
    "..................XNkl:;;:.:::oXXXXXXXXXXXXXXXXX0:'.NXXXXXXXXXXXXXXXNkc:cccc:..:d0..................\n",
    "......................NKko.;;;:lxOKX.XXXXXXXXXXXXNN.XXXXXXXXXXX..X0Odc::::cdOX......................\n",
    "...........................N0kdl:;;:;;:clodxxkkkkOO.Okkxxddolc::::;:cldOK...........................\n",
    "..................................XKOkxdolcc:;;::;;.::;::cllodxk0KN.................................\n",
    "....................................................................................................\n",
    "\"\"\")) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11468451846417028993973305727890751485"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_grid_paths(Grid(r\"\"\"\n",
    ".....................................................................................................\n",
    ".................WXK0kxdd.oooooooooodxOXW............................................................\n",
    "..........W0xdoooooooodxk.00KKKKK00OxdoolokXW........................................................\n",
    "..........Wk',xKXNW....................WXkollOW......................................................\n",
    "............0:c0W..........WNN.............NkccOW....................................................\n",
    ".............Xd:dN.........Nkooodk0XW........Nk:lKW..................................................\n",
    "..............W0lckN........WNKOxooooodk0NW....Xo:ckW................................................\n",
    "................WOccxX.............WX0xdoooodOKNWkc;dN.WKOK..........................................\n",
    "..................WOocoON................WXOxoododd:.ld:..cX.........................................\n",
    ".....................XxlloxKN..................NOl:'....;dKW.........................................\n",
    ".......................WXkoloooxO0XNNWWWNXK0k.oll;'...;ON............................................\n",
    "...........................WN0kdooooooooooooo.kKO;...dN......WN0kdocc.WW.clodk0NW....................\n",
    "....................NX0OkkkO0XWW...WWNNNW......Xc...xW...WXko;.....',.Nd.;'.....;lkKW................\n",
    "...............WXOo.;;::ccc::::cokKW..........Wx...oN..Nkc...,cdOKXNW....WNXKOdl;...;o0N.............\n",
    ".............Nkc,;c.OXW.....WX0xl:;:lkXW......X;.,d0.Xd'..;d0N..................WKkl'..,dKW..........\n",
    "..........WKo,,lONW..............WXkl:;cxKW......oNXd'..cON.........................NOl...lKW........\n",
    ".........Kl',xX......................W0dc::lkKo..cd;..c0W.............................WXd'..oX.......\n",
    ".......Nd''xN...........................WXOoc:'.....'kW..................................Xo..,.W.....\n",
    "......K:.lK.................................WNx'...;K.....................................W0;..xW....\n",
    "....W0,.xN..................................Nx...o0X........................................X:..xW...\n",
    "....0,.kW...................................WOcl.W...........................................X:.'O...\n",
    "...K;.xW.........................................................................................cN..\n",
    "..Nl.lN.......................................................................................Wl..k..\n",
    "..k.,K.........................................................................................k..oW.\n",
    ".Nc.oW............................................................................................cN.\n",
    ".K,.O..........................................................................................K,.:X.\n",
    ".k.;X..........................................................................................K,.:N.\n",
    ".x.:N..........................................................................................0'.lW.\n",
    ".d.cN..........................................................................................k.....\n",
    ".d.cN.........................................................................................Wo..O..\n",
    ".x.;X.........................................................................................K,.:X..\n",
    ".O.'0........................................................................................Wx..dW..\n",
    ".K,.x........................................................................................X;.,K...\n",
    ".Wl.cN......................................................................................Wd..dW...\n",
    "..O.'O......................................................................................0,.:X....\n",
    "..Nc.lN....................................................................................Nl..O.....\n",
    "...O'.k...................................................................................Wx..oW.....\n",
    "...Wd.;K..................................................................................0,.:X......\n",
    "....Xc.cN................................................................................X:.'0.......\n",
    ".....K,.oN..............................................................................No..xW.......\n",
    "......0,.dN............................................................................Wx..oW........\n",
    ".......0,.oN..............................................................................cN.........\n",
    "........0;.cX.........................................................................0,.;K..........\n",
    ".........Kc.;0W......................................................................X;..0...........\n",
    "..........Nd..dN....................................................................Xc...............\n",
    "...........W0;.;OW.................................................................Xc..kW............\n",
    ".............Nd'.c0W..............................................................K:..kW.............\n",
    "...............Xo..:ON..........................................................Nk'.;0W..............\n",
    ".................Xd'.,dKW.....................................................NO:..oX................\n",
    "...................Xx:..:d0N........................WNXK0000KKXNW.........WXO....c0W.................\n",
    ".....................WKx:'.,cdOKNW...........WNKOxol;'........'',;:cclllc.;......KW..................\n",
    "........................WXOo:,..';cloodddoolc;'..',;ldk00K000Okxoolc:::::.ldOXW......................\n",
    ".............................NKOxolc:;;,,;:ccodk0NWW.................................................\n",
    ".....................................WWWWW...........................................................\n",
    ".....................................................................................................\n",
    "\"\"\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can you verify that these last three answers are correct?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Paths in a 2 x *n* Grid\n",
    "\n",
    "Nicolas Schank proposes this problem:\n",
    "\n",
    "> How many ways can you arrange the integers 1 through 2*n* in a 2 x *n* grid such that any two consecutive numbers are placed into squares that share a side?\n",
    "\n",
    "Another way of stating the problem is: how many paths are there that visit every square in  2 x *n* grid once,  moving from a square to any of its orthogonal neighbors. (The integers 1 is then the first square in the path, 2 the second, and so on.)\n",
    "\n",
    "The **brute force enumeration** strategy would be to generate all (2*n*)! permutations of the integers and check if they form a valid path. That should work fine up to  *n* = 6, but not much beyond that.\n",
    "\n",
    "Therefore I'll go to the  **incremental enumeration strategy**, in which I start with paths of length 1 and extend them one square at a time, checking for validity at each step, until we find all the paths that reach all 2*n* squares. (This is not an **abstract** strategy: these are individual paths, not summaries of them.)\n",
    "\n",
    "I'll describe a `Path` with three components, `Path(end, n, squares)`:\n",
    "  - `end`: the `(x, y)` coordinates of the square that the path ends in.\n",
    "  - `n`: the width of the grid. (This is the same for every path in a problem instance, but I had to store it somewhere.)\n",
    "  - `squares`: a dict of `{(x, y): i}` entries giving the order of the squares visited.\n",
    "  \n",
    "The approach then is that we start with one-square paths of the form `Path(s, n, {s: 1})` for every square `s` in the grid. Then we `iterate` calling `extend_paths` until we have all the complete paths. A path is extended by  placing the integer `len(squares) + 1` in any square that neighbors `path.end` and does not already appear in `path.squares`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "Path = namedtuple('Path', 'end, n, squares')\n",
    "\n",
    "def incremental_count_paths(n) -> [Path]:\n",
    "    \"\"\"Extend every path as far as possible and return the ones of length 2*n\"\"\"\n",
    "    grid = product(range(n), (0, 1))\n",
    "    paths = [Path(s, n, {s: 1}) for s in grid]\n",
    "    return iterate(extend_paths, paths, 2 * n - 1)\n",
    "\n",
    "def extend_paths(paths) -> [Path]:\n",
    "    \"\"\"Return a list of paths that validly extend these paths by one square.\"\"\"\n",
    "    return [Path(end2, n, {end2: len(squares) + 1, **squares})\n",
    "            for (end,  n, squares) in paths\n",
    "            for end2 in neighbors(n, *end) if end2 not in squares]\n",
    "\n",
    "def neighbors(n, x, y):\n",
    "    \"\"\"The squares that neighbor `(x, y)` in a nx2 grid.\"\"\"\n",
    "    if x < n-1: yield (x + 1, y)\n",
    "    if x > 0:   yield (x - 1, y)\n",
    "    yield (x, 1 - y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Path(end=(0, 1), n=2, squares={(0, 1): 4, (1, 1): 3, (1, 0): 2, (0, 0): 1}),\n",
       " Path(end=(1, 0), n=2, squares={(1, 0): 4, (1, 1): 3, (0, 1): 2, (0, 0): 1}),\n",
       " Path(end=(0, 0), n=2, squares={(0, 0): 4, (1, 0): 3, (1, 1): 2, (0, 1): 1}),\n",
       " Path(end=(1, 1), n=2, squares={(1, 1): 4, (1, 0): 3, (0, 0): 2, (0, 1): 1}),\n",
       " Path(end=(1, 1), n=2, squares={(1, 1): 4, (0, 1): 3, (0, 0): 2, (1, 0): 1}),\n",
       " Path(end=(0, 0), n=2, squares={(0, 0): 4, (0, 1): 3, (1, 1): 2, (1, 0): 1}),\n",
       " Path(end=(1, 0), n=2, squares={(1, 0): 4, (0, 0): 3, (0, 1): 2, (1, 1): 1}),\n",
       " Path(end=(0, 1), n=2, squares={(0, 1): 4, (0, 0): 3, (1, 0): 2, (1, 1): 1})]"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "incremental_count_paths(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try to get a better understanding by visualizing the paths:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "def show_paths(paths, cols=4):\n",
    "    \"\"\"Plot all the paths in a matrix of subplots.\"\"\"\n",
    "    n = paths[0].n\n",
    "    P = len(paths)\n",
    "    fig, axs = plt.subplots(P // cols, cols, figsize=(14, 2 * n))\n",
    "    for y in range(P // cols):\n",
    "        for x in range(cols):\n",
    "            if paths:\n",
    "                path = paths.pop()\n",
    "                _, X, Y = zip(*(sorted((i, x, y) for (x, y), i in path.squares.items())))\n",
    "                axs[y, x].axis('off')\n",
    "                axs[y, x].plot(X, Y, 's-', clip_on=False)\n",
    "                axs[y, x].plot(X[:1], Y[:1], 'rs', clip_on=False)\n",
    "    fig.tight_layout(pad=3.0)\n",
    "    fig.suptitle(f'{P} Paths for a 2x{n} Grid:')\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1008x288 with 8 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1008x432 with 16 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1008x576 with 28 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1008x720 with 44 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1008x864 with 64 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for n in (2, 3, 4, 5, 6):\n",
    "    show_paths(incremental_count_paths(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's also create a table of number of paths as a function of `n`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 2,\n",
       " 2: 8,\n",
       " 3: 16,\n",
       " 4: 28,\n",
       " 5: 44,\n",
       " 6: 64,\n",
       " 7: 88,\n",
       " 8: 116,\n",
       " 9: 148,\n",
       " 10: 184,\n",
       " 11: 224,\n",
       " 12: 268}"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{n: len(incremental_count_paths(n)) for n in range(1, 13)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The visualization was interesting, and the number of valid paths makes it clear that this is not growing too quickly; probably polynomial rather than exponential. But I have no clue about a general formula for *n*.\n",
    "\n",
    "I'll try the **standing on shoulders** approach. I'll search for the first few elements of the sequence,\n",
    "[\"2, 8, 16, 28, 44, 64\"](https://www.google.com/search?q=%222%2C+8%2C+16%2C+28%2C+44%2C+64%22), and see if anyone has reported on them.  It turns out the [first search result](https://oeis.org/search?q=2%2C+8%2C+16%2C+28%2C+44%2C+64&language=english&go=Search) is from the online encyclopedia of integer sequences (a famous source for this kind of knowledge) for a sequence described as \"Number of (directed) Hamiltonian paths in the n-ladder graph.\" I know that a Hamiltonian path is a path that visits each vertex once, so that sounds right, and I had never heard the phrase \"n-ladder graph,\" but it makes sense that it is a 2 x *n* grid. So it looks like we're in the right place. The page gives this **calculation** formula:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_paths(n): return 2 if n == 1 else 2 * (n ** 2 - n + 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all(calculate_paths(n) == len(incremental_count_paths(n)) for n in range(1, 13))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our work here is done."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Counting Positions in Fischerandom Chess\n",
    "\n",
    "> In this [variant](https://en.wikipedia.org/wiki/Chess960) of chess, the pieces are set up in a random but restricted fashion. The pawns are in their regular positions, and the major white pieces are placed randomly on the first rank, with two restrictions: the bishops must be placed on opposite-color squares, and the king must be placed between the rooks. The black pieces are set up to mirror the white pieces. How many starting positions are there?\n",
    "\n",
    "We can answer by **enumerate and test**: generate all *distinct* permutations of the pieces and count the number that are valid:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "def valid_position(pieces) -> bool:\n",
    "    \"\"\"Valid if bishops are on different colors, and king is between rooks.\"\"\"\n",
    "    pieces = ''.join(pieces) # make `pieces` be a string (e.g. 'RNKBRQBN')\n",
    "    B, R, K = map(pieces.index, 'BRK')\n",
    "    b, r    = map(pieces.rindex, 'BR')\n",
    "    return (color(B) != color(b)) and (r < K < R or R < K < r)\n",
    "\n",
    "def color(square): return 'BW'[square % 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "960"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quantify(set(permutations('RNBKQBNR')), valid_position)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(*Note:* initially my program failed because I forgot that while tuples, lists and strings all have an `index` method, only strings have `rindex`.  How annoying! I had to fix the problem by adding `pieces = ''.join(pieces)`. In Ruby, both strings and arrays have `index` and `rindex`. In Julia, both stringsss and arrays have `findfirst` and `findlast`. In  Java and Javascript, both strings and lists/arrays have both `indexOf` and `lastIndexOf`. What's wrong with Python? )\n",
    "\n",
    "We could also do this by **calculation**. Let's handle the bishops first. The first bishop can go on any of the 8 squares, but then the second bishop has to go on an opposite color, so that's 4 choices. But then we divide by 2! = 2 because the two bishops are indistinguishable. Next, place the other pieces in the 6 remaining squares, for 6! possibilities, but divide by 2! because the knights are indistinguishable, and divide by 3! because, out of the 3! ways of ordering R-K-R left-to-right, only 2 of them are valid, but the 2 cancels out because the rooks are indistinguishable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "960.0"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(8 * 4 / factorial(2))   *   factorial(6) / (factorial(2) * factorial(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Counting Change\n",
    "\n",
    "> How many ways are there to select coins that add up to a specified amount of money? For example, to make 10 cents change with  10, 5, and 1 cent coins, there are four ways: `{10,  5+5,  5+1+1+1+1+1,  1+1+1+1+1+1+1+1+1+1}`. \n",
    "\n",
    "This is a well-known problem with a [Wikipedia page](https://en.wikipedia.org/wiki/Change-making_problem). But we'll tackle it in our own way. To start, I will use the term **mint** to describe a set of coin denominations. (I could have used  **denominations**, but \"mint\" is shorter, and I can say \"mints\" to refer to several sets,  but I can't say \"denominationses.\") The US mint produces coins in the following denominations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "mint = (100, 50, 25, 10, 5, 1) # Denominations of coins in US"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conceptually a mint is a set, but the counting will be more efficient if the mint is ordered, with the largest values first.\n",
    "\n",
    "For this problem we will take a recursive **divide and conquer** approach with **remembering**. `count_change(amount, mint)` says how many ways there are to add up to `amount` using coins in `mint`. For example, `count_change(10, (10, 5, 1))` is 4. We analyze `count_change` as follows:\n",
    "- There are three simple cases where we can answer immediately without having to divide the input:\n",
    "   - If the amount is zero cents there is one way to add up to the amount: with no coins.\n",
    "   - If the amount is negative, there are no ways.\n",
    "   - If the amount is positive but there are no denominations in the mint then there are no ways.\n",
    "- Otherwise, we **divide** the possibilities into two parts and **conquer** by adding up the numbers from each part:\n",
    "   - Part 1: Use a coin of the first denomination in the mint. Figure out how many ways to make the rest of the amount with the mint.\n",
    "   - Part 2: Skip the first denomination in the mint. Figure out how many ways to make the whole amount without the denomination."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "@lru_cache(None)\n",
    "def count_change(amount, mint=mint) -> int:\n",
    "    \"\"\"The number of ways of adding up to `amount`, using coins from `mint`.\"\"\"\n",
    "    return (1 if amount == 0 else\n",
    "            0 if amount < 0 or not mint else\n",
    "            count_change(amount - mint[0], mint) + \n",
    "            count_change(amount, mint[1:]))                "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can count the ways to make change for various amounts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_change(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_change(25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "252"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_change(99)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How many different piles of coins total up to $1000?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13398445413854501"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_change(10**5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Aside: Recursion and @lru_cache\n",
    "\n",
    "*Quiz Question*: Wait a minute! Why doesn't `count_change(10**5)` raise a `RecursionError`? Doesn't `count_change` have to follow a chain of 100,000 recursive calls to handle the case where 100,000 1's total up to the amount? And isn't 100,000 larger than `sys.getrecursionlimit()`?\n",
    "    \n",
    "*Quiz Answer*: We would indeed get a `RecursionError` if any of the following were true:\n",
    "- The `@lru_cache` decorator were not used on `count_change`.\n",
    "- We called `count_change(10**6)` instead of `count_change(10**5)`.\n",
    "- The `mint` was ordered with the `1` first, not the `100`.\n",
    "- The order of the two recursive calls to `count_change` were reversed. \n",
    "\n",
    "As it is, `count_change` fills the cache in such a way that it avoids too many recursive calls. To see what's happening, I'll create a plot of the call depth (on the y axis) for each successive recursive call to `count_change` (on the x-axis) in the execution of `count_change(10**5)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "@lru_cache(None)\n",
    "def count_change_with_depths(amount, mint=mint) -> int:\n",
    "    \"\"\"The number of ways of adding up to `amount`, using coins from `mint`.\n",
    "    This version appends call depths to `plot_depths.depths`.\"\"\"\n",
    "    global depth, depths\n",
    "    depth += 1\n",
    "    depths.append(depth)\n",
    "    result = (1 if amount == 0 else\n",
    "              0 if amount < 0 or not mint else\n",
    "              count_change_with_depths(amount - mint[0], mint) + \n",
    "              count_change_with_depths(amount, mint[1:]))\n",
    "    depth -= 1\n",
    "    return result\n",
    "\n",
    "def plot_depths(amount, mint=mint, show=slice(None)):\n",
    "    \"\"\"Plot the call depths for `count_change(amount, mint)`.\"\"\"\n",
    "    global depth, depths\n",
    "    depth, depths = 0, []\n",
    "    count_change_with_depths.cache_clear()\n",
    "    count_change_with_depths(amount, mint)\n",
    "    plt.figure(figsize=(12, 6))\n",
    "    plt.xlabel('Call Number'); plt.ylabel('Depth')\n",
    "    X, Y = range(1, len(depths) + 1), depths\n",
    "    plt.plot(X[show], Y[show], '.-')\n",
    "    plt.grid(True); plt.gca().invert_yaxis()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(Note: this would be cleaner if done as a decorator to track depths, and with `depth` and `depths` being attributes rather than global variables.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_depths(10**5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It  looks like we first do a little over 1000 levels of recursion, then start popping the stack to return to the top level. But we can't really see from this plot how it happens; let's zoom in on the key part of the graph:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_depths(10**5, show=slice(997, 1130))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that right at call number 1000 is the first time when the depth does not increase; it goes sideways for one step. Then we recurse to a maximum depth of 1014, and then gradually start zig-zagging back to the top level. We can figure out why this happens:\n",
    "- First there are 1000 consecutive calls to `count_change_depths(amount - 100, mint)`. We can see there is a blue dot right at the (1000 calls, 1000 depth) grid point.\n",
    "- The last of these calls is a call to `count_change_depths(0, mint)`, which returns without a recursive call, giving us the first sideways step (at  call number 1001-1002, depth 1001).\n",
    "- Call 1002 is the first recursive call to `count_change_depths(amount, mint[1:])` (this time with `amount=100`).\n",
    "- We eventually fill in the cache entries for all amounts below 100, without ever going below depth 1014.\n",
    "- We start returning up the stack: a zig up.\n",
    "- Eventually we return and move on to a recursive call of `count_change_depths(200, mint[1:])`. \n",
    "- To fill in the cache entries for 200 to 100: a zig down, but never as far down as 2014.\n",
    "- We alternate returning up the stack, and going deeper to fill in entries for the next 100 amounts."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The pattern is easier to see with a limited mint of three coins: `(100, 10, 1)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_depths(10**5, mint=(100, 10, 1), show=slice(988, 3314))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each parallelogram corresponds to another 100 amounts being filled in. The depth never goes below 1021, and the parallelograms are gradually making their way up from the depths to the surface."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variant Problem: Show Change\n",
    "\n",
    "Instead of *counting* how many ways there are to make change, we could *show* the actual coins in each way. \n",
    "I'll use the term **purse** for a collection of coins, because it is a [**bag**](https://en.wikipedia.org/wiki/Multiset) of coins. I'll implement `Purse` as a `Counter` of `{denomination: count}` pairs. (A purse is different than a mint: a mint just lists the denominations; a purse lists the denominations and how many coins we have of each denomination.) For example, `Purse({10: 2, 1: 4})` means two 10-cent coins and four 1-cent coins, for a total of 24 cents. I'll give the `Purse` class two methods to add or subtract one or more coins of the same denomination; these methods produce a new `Purse` and do not modify the old one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Purse(Counter): \n",
    "    \"\"\"A bag of coins (or of any objects, really).\"\"\"\n",
    "    def add(self, coin, n=1): return Purse(self + Counter({coin: n}))\n",
    "    def sub(self, coin, n=1): return Purse(self - Counter({coin: n}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So `show_change` will return a list of Purses. The overall structure of the function is the same as `count_change`, but because of the additional complication of having to build up lists of purses for the results, I switch the body of the function from an *expression* to *statements*, to allow for the assignment of intermediate values to mnemonic variable names."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [],
   "source": [
    "@lru_cache(None)\n",
    "def show_change(amount, mint=mint) -> [Purse]:\n",
    "    \"\"\"List all the purses that adds up to `amount`, using `mint`.\"\"\"\n",
    "    if amount == 0:\n",
    "        return [Purse()] \n",
    "    elif amount < 0 or not mint:\n",
    "        return []\n",
    "    else:\n",
    "        coin = mint[0]\n",
    "        use  = show_change(amount - coin, mint)\n",
    "        skip = show_change(amount, mint[1:])\n",
    "        return [purse.add(coin) for purse in use] + skip"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Purse({1: 1, 10: 1}),\n",
       " Purse({1: 1, 5: 2}),\n",
       " Purse({1: 6, 5: 1}),\n",
       " Purse({1: 11})]"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_change(11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Purse({5: 1, 10: 1}),\n",
       " Purse({1: 5, 10: 1}),\n",
       " Purse({5: 3}),\n",
       " Purse({1: 5, 5: 2}),\n",
       " Purse({1: 10, 5: 1}),\n",
       " Purse({1: 15})]"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_change(15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Purse({25: 1}),\n",
       " Purse({5: 1, 10: 2}),\n",
       " Purse({1: 5, 10: 2}),\n",
       " Purse({5: 3, 10: 1}),\n",
       " Purse({1: 5, 5: 2, 10: 1}),\n",
       " Purse({1: 10, 5: 1, 10: 1}),\n",
       " Purse({1: 15, 10: 1}),\n",
       " Purse({5: 5}),\n",
       " Purse({1: 5, 5: 4}),\n",
       " Purse({1: 10, 5: 3}),\n",
       " Purse({1: 15, 5: 2}),\n",
       " Purse({1: 20, 5: 1}),\n",
       " Purse({1: 25})]"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_change(25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Variant Problem: Show Limited Change\n",
    "\n",
    "> The above assumed that we had a limitless supply of coins of each denomination. What if we only have a limited number of coins in our purse?\n",
    "\n",
    "I'll define `show_limited_change`, which, instead of takling a mint as argument, takes a purse with a specific number of coins of each denomination. We use the same strategy as `show_change`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "def show_limited_change(amount, purse) -> [Counter]:\n",
    "    \"\"\"List all the ways of making change that adds up to `amount`, using `coins`.\"\"\"\n",
    "    if amount == 0:\n",
    "        return [Purse()] \n",
    "    elif amount < 0 or not purse:\n",
    "        return []\n",
    "    else:\n",
    "        coin = max(purse)\n",
    "        use  = show_limited_change(amount - coin, purse.sub(coin, n=1))\n",
    "        skip = show_limited_change(amount,        purse.sub(coin, n=purse[coin]))\n",
    "        return [purse.add(coin) for purse in use] + skip"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Purse({1: 1, 10: 1}),\n",
       " Purse({1: 1, 5: 2}),\n",
       " Purse({1: 6, 5: 1}),\n",
       " Purse({1: 11})]"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_limited_change(11, Purse({10: 4, 5: 3, 1: 11}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Purse({1: 1, 10: 1})]"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_limited_change(11, Purse({10: 4, 5: 1, 1: 4}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Purse({25: 1})]"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_limited_change(25, Purse({25: 1, 10: 1, 5: 2, 1: 4})) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_limited_change(25, Purse({10: 12, 1: 4}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variant Problem: Optimal mint\n",
    "\n",
    "The  [July 20, 2018 Riddler](https://fivethirtyeight.com/features/damn-the-torpedoes-two-puzzles-ahead/) poses this problem (slightly edited  here):\n",
    "\n",
    "> If Riddler Nation needed to make change (anywhere from 0.01 to 0.99) and was establishing its own mint, what values of coins would be ideal to yield the smallest number of coins in an average transaction? You can assume that all amounts of change from 0.01 to 0.99 are equally likely. Let’s limit our mint to four different coin denominations.  \n",
    "\n",
    "Technically this is an optimization problem, not a counting problem, but we'll answer it anyway. (It is an interesting problem that boasts at least one  [journal article](https://cs.uwaterloo.ca/~shallit/Papers/change2.pdf).) Here's how I address the problem:\n",
    "- The function `meancoins(mint)` will give the mean number of coins required to make each amount of change from 0 to 99 cents.\n",
    "- The function `mincoins(amount, mint)` computes how many coins are needed for a given amount, or returns an absurdly large number (`maxsize`) if the amount cannot be made. (A mint that contains a `1` can make any amount.)\n",
    "- The variable `mints` holds a list of possible four-coin mints, such as `(27, 13, 3, 1)`.  I know that a 1 cent piece is required; otherwise I can't make an amount of 1 cent. That leaves 3 coins that could be anywhere from 2 to 99 cents; `mints` enumerate all the possible combinations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "152096"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@lru_cache(None)\n",
    "def mincoins(amount, mint) -> int:\n",
    "    \"\"\"The minimum number of coins, taken from mint, that add to amount.\"\"\"\n",
    "    return (0       if amount == 0 else\n",
    "            maxsize if not mint or amount < min(mint) else\n",
    "            min(mincoins(amount, mint[1:]),\n",
    "                mincoins(amount - mint[0], mint) + 1))\n",
    "\n",
    "def meancoins(mint, minimizer=mincoins, amounts=range(100)) -> float: \n",
    "    \"\"\"The mean number of coins needed to make change for all the amounts.\"\"\"\n",
    "    return sum(minimizer(a, mint) for a in amounts) / len(amounts)\n",
    "\n",
    "mints = [(L, M, S, 1) for S, M, L in combinations(range(2, 100), 3)]\n",
    "len(mints)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now I can sort the mints by `meancoins`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 33.2 s, sys: 2.42 s, total: 35.7 s\n",
      "Wall time: 49.1 s\n"
     ]
    }
   ],
   "source": [
    "%time mints.sort(key=meancoins)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And look at the top 10, along with the US system:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [],
   "source": [
    "topmints = mints[:10] + [(25, 10, 5, 1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(25, 18, 5, 1): 3.89,\n",
       " (29, 18, 5, 1): 3.89,\n",
       " (30, 18, 4, 1): 3.9,\n",
       " (28, 17, 4, 1): 3.91,\n",
       " (29, 19, 4, 1): 3.91,\n",
       " (30, 19, 5, 1): 3.91,\n",
       " (28, 21, 5, 1): 3.91,\n",
       " (32, 19, 4, 1): 3.92,\n",
       " (30, 23, 5, 1): 3.92,\n",
       " (31, 14, 6, 1): 3.92,\n",
       " (25, 10, 5, 1): 4.7}"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{mint: meancoins(mint) for mint in topmints}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interesting! The mint on the top line, (25, 18, 5, 1), is almost the US system of coins; we just need to trade in the dime for an 18 cent piece. It takes an average of 3.89 coins to make any amount from 0 to 99 (as does the mint on the second line, `(29, 18, 5, 1)`). We could also consider trading in the quarter for a 29 or 30 cent piece. The US system, at 4.7 mean number of coins, is almost a full coin behind the best mint.\n",
    "\n",
    "However, I'm not sure that `meancoins` is the best measure of a mint system. For one thing, it can be mentally taxing to compute the minimum purse of coins for an amount. It is mentally easier to use a **greedy approach** which says: to make change, start with the largest coin available, and use as many of those as possible; then continue to the other coins in decreasing order. With this approach you never have to compare two possible options. \n",
    "\n",
    "I'll define `greedy_mincoins(amount, mint)` to say how many coins are needed to make `amount` from `mint` using the greedy aprroach, and  `show_greedy_limited_mincoins(amount, purse)` to show the coins that can make up `amount`, drawing only from `purse`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [],
   "source": [
    "def greedy_mincoins(amount, mint) -> int: \n",
    "    \"\"\"The number of ways of adding up to `amount`, using coins from `mint`.\"\"\"\n",
    "    bank = Purse({d: 1000 for d in mint})\n",
    "    return total(show_greedy_limited_mincoins(amount, bank))\n",
    "\n",
    "def show_greedy_limited_mincoins(amount, purse) -> Purse:\n",
    "    \"\"\"Change, taken greedily from purse, that adds to amount.\"\"\"\n",
    "    change = Purse()\n",
    "    for coin in sorted(purse, reverse=True):\n",
    "        while coin <= amount:\n",
    "            amount -= coin\n",
    "            change[coin] += 1\n",
    "            purse = purse.sub(coin)\n",
    "    return change if amount == 0 else None\n",
    "\n",
    "def is_canonical(mint) -> bool:\n",
    "    \"\"\"Does this mint give the same results with `mincoins` and `greedy_mincoins`?\"\"\"\n",
    "    return same(lambda a: mincoins(a, mint), \n",
    "                lambda a: greedy_mincoins(a, mint), \n",
    "                range(100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The greedy approach is not optimal for all amounts and mints. For example, to make the amount 30 with the mint `(25, 10, 1)` the greedy approach takes the 25, and then must add five 1-cent coins for a total of 6 coins. The optimal `mincoins` approach selects three 10-cent coins. A mint for which the greedy algorithm is optimal for all amounts is called a **canonical** system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "greedy_mincoins(30, (25, 10, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "greedy_mincoins(6, (4, 3, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Purse({25: 1, 1: 5})"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_greedy_limited_mincoins(30, Purse({25: 1, 10: 3, 1: 7}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's compare the mean number of coins needed under the optimal and greedy approaches, and check which mints are canonical:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(25, 18, 5, 1): (3.89, 4.58, False),\n",
       " (29, 18, 5, 1): (3.89, 4.45, False),\n",
       " (30, 18, 4, 1): (3.9, 4.38, False),\n",
       " (28, 17, 4, 1): (3.91, 4.44, False),\n",
       " (29, 19, 4, 1): (3.91, 4.41, False),\n",
       " (30, 19, 5, 1): (3.91, 4.48, False),\n",
       " (28, 21, 5, 1): (3.91, 4.62, False),\n",
       " (32, 19, 4, 1): (3.92, 4.41, False),\n",
       " (30, 23, 5, 1): (3.92, 4.6, False),\n",
       " (31, 14, 6, 1): (3.92, 4.45, False),\n",
       " (25, 10, 5, 1): (4.7, 4.7, True)}"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{mint: (meancoins(mint, mincoins), meancoins(mint, greedy_mincoins), is_canonical(mint))\n",
    " for mint in topmints}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now it looks like maybe `(30, 18, 4, 1)` is the best mint: it is only 0.01 behind the leader in optimal score, and it has the best greedy score.\n",
    "\n",
    "We also see that among the mints shown, only the US system, `(25, 10, 5, 1)` is canonical: it gets the same score under greedy and optimal approaches. However, even its optimal score,  4.7, is worse than the greedy score of all the mints above it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Variant: Mean Purse Size\n",
    "\n",
    "Here's another criteria for a mint system: I would like to not accumulate too many coins in my purse/pocket. Suppose I do a cash transaction in which I owe a certain amount of cents, `amount`. If I happen to have coins in my purse that add up to `amount`, I will pay with them (and end up with fewer coins). If I don't, I'll pay for the cents with a dollar bill, and I'll receive `100 - amount` in change back (and end up with more coins). (I won't allow the scenario where, say, I owe 24¢ and pay 25¢ and get 1¢ back.) Will some mints lead to accumulation of more coins than others? I'll do a random **simulation** to see:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [],
   "source": [
    "@lru_cache(None)\n",
    "def purse_stats(mint, t=25000, seed=42):\n",
    "    \"\"\"The mean and standard deviation of the number of coins in a purse\n",
    "    after each of `t` random transactions.\"\"\"\n",
    "    random.seed(seed)\n",
    "    purse = Purse()\n",
    "    bank  = Purse({c: 100 for c in mint})\n",
    "    sizes = []\n",
    "    for _ in range(t):\n",
    "        amount = random.randrange(1, 100)\n",
    "        pay    = show_greedy_limited_mincoins(amount, purse)\n",
    "        if pay:\n",
    "            purse -= pay\n",
    "        else:\n",
    "            change = show_greedy_limited_mincoins(100 - amount, bank)\n",
    "            purse += change\n",
    "        sizes.append(total(purse))\n",
    "    return mean(sizes), stdev(sizes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I can compare various mints:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5.6928, 4.441767365800834)"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "purse_stats((25, 10, 5, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(6.47408, 4.9299959784987575)"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "purse_stats((25, 18, 5, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This says that the US system leaves me, on average, with fewer coins in my purse. But the standard deviation is large compared to the mean, so this may not be very reliable in the short run.\n",
    "\n",
    "One more thing I'm interested in is what is the *worst* amount for a mint: the amount that requires the most coins? And what are those coins?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "def worst_amount(mint):\n",
    "    \"\"\"What amount (and coins) requires the most number of coins for mint?\"\"\"\n",
    "    amount = max(range(100), key=lambda a: mincoins(a, mint)) # Worst amount\n",
    "    coins = min((list(c.elements()) for c in show_change(amount, mint)), key=len)\n",
    "    return amount, coins\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(94, [1, 1, 1, 1, 5, 10, 25, 50])"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "worst_amount(mint)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a report summarizing all we have learned about the various mints:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mint_report(mints):\n",
    "    print('Mint of Coins   Mean Greedy Canon  PurseSize  Amount-requiring-most-coins')\n",
    "    for mint in mints:\n",
    "        m, g = meancoins(mint), meancoins(mint, greedy_mincoins)\n",
    "        mu, sd = purse_stats(mint)\n",
    "        a, coins = worst_amount(mint)\n",
    "        print(f'{mint}  {m:.2f}  {g:.2f}  {m==g!s:5} {mu:4.1f} ± {sd:3.1f}  {a:2}¢ {len(coins)}: {coins[::-1]}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mint of Coins   Mean Greedy Canon  PurseSize  Amount-requiring-most-coins\n",
      "(25, 18, 5, 1)  3.89  4.58  False  6.5 ± 4.9  14¢ 6: [5, 5, 1, 1, 1, 1]\n",
      "(29, 18, 5, 1)  3.89  4.45  False  8.4 ± 6.5  14¢ 6: [5, 5, 1, 1, 1, 1]\n",
      "(30, 18, 4, 1)  3.90  4.38  False  7.1 ± 5.7  15¢ 6: [4, 4, 4, 1, 1, 1]\n",
      "(28, 17, 4, 1)  3.91  4.44  False  6.4 ± 5.0  99¢ 7: [28, 28, 17, 17, 4, 4, 1]\n",
      "(29, 19, 4, 1)  3.91  4.41  False  7.7 ± 6.0  15¢ 6: [4, 4, 4, 1, 1, 1]\n",
      "(30, 19, 5, 1)  3.91  4.48  False  6.1 ± 4.6  14¢ 6: [5, 5, 1, 1, 1, 1]\n",
      "(28, 21, 5, 1)  3.91  4.62  False  7.6 ± 6.1  19¢ 7: [5, 5, 5, 1, 1, 1, 1]\n",
      "(32, 19, 4, 1)  3.92  4.41  False  6.5 ± 5.2  15¢ 6: [4, 4, 4, 1, 1, 1]\n",
      "(30, 23, 5, 1)  3.92  4.60  False  6.4 ± 4.9  19¢ 7: [5, 5, 5, 1, 1, 1, 1]\n",
      "(31, 14, 6, 1)  3.92  4.45  False  6.8 ± 5.1  98¢ 7: [31, 31, 14, 14, 6, 1, 1]\n",
      "(25, 10, 5, 1)  4.70  4.70  True   5.7 ± 4.4  94¢ 9: [25, 25, 25, 10, 5, 1, 1, 1, 1]\n"
     ]
    }
   ],
   "source": [
    "mint_report(topmints)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The US system (bottom row) has the advantage of being canonical, and of being natural for people with five fingers on a hand. But its mean number of coins needed is high, as is its maximum number of coins, 9 (to make 94¢). The `(25, 18, 5, 1)` mint has the best (tied) `mincoins` mean score, the best (tied) maximum-number-of-coins score, and pretty good greedy and purse-size scores. It might have been a better system overall, but not by enough of a margin to seriously consider a switch."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem: Drink Gift Cards\n",
    "\n",
    "[538 Riddler](https://fivethirtyeight.com/features/does-your-gift-card-still-have-free-drinks-on-it/) presents this problem:\n",
    "    \n",
    "> Lucky you! You’ve won two gift cards, each loaded with 50 free drinks from your favorite coffee shop, Riddler Caffei-Nation. The cards look identical, and because you’re not one for record-keeping, you randomly pick one of the cards to pay with each time you get a drink. One day, the clerk tells you that the card you presented doesn’t have any drink credits left on it. How many free drinks can you expect are still available on the other card?\n",
    "\n",
    "Can I **enumerate** all sequences of choosing one card or the other? No: that would be (100 choose 50) ≅ 10<sup>29</sup> sequences. But I can to use recursive **divide and conquer**:  I'll define `other_card(a, b)` to return a probability distribution of the number of drinks remaining on the \"other\" card, when we start with two cards with `a` and `b` drinks remaining, respectively, and we use them until one card is exhausted. At every step, the function considers decrementing both `a` and `b`, with equal probability (1/2 each). \n",
    "\n",
    "I can define a probability distribution, `Dist`, as a subclass of `Counter` to which I add a method for multiplying by a scalar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Dist(Counter):\n",
    "    \"\"\"A frequency distribution of {item: frequency}.\"\"\"\n",
    "    def __mul__(self, scalar): return Dist({x: scalar * self[x] for x in self})\n",
    "    __rmul__ = __mul__\n",
    "\n",
    "@lru_cache(None)\n",
    "def other_card(a, b):\n",
    "    \"\"\"Probability distribution of drinks remaining on other card when one card runs out.\"\"\"\n",
    "    a, b = sorted((a, b)) # Ensure a <= b\n",
    "    return (Dist({b: 1})  if a == 0 else\n",
    "            Dist(1/2 * other_card(a - 1, b) + \n",
    "                 1/2 * other_card(a, b - 1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If I have one free drink on each card and I use one, then the other card has one drink with probability  1.0:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dist({1: 1.0})"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "other_card(1, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If I start with two free drinks on each card and I use them randomly until one runs out, then it is equally likely that there are 1 or 2 left on the other card:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dist({2: 0.5, 1: 0.5})"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "other_card(2, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the full probability distribution for the original question:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dist({50: 1.7763568394002505e-15,\n",
       "      49: 4.440892098500626e-14,\n",
       "      48: 5.662137425588298e-13,\n",
       "      47: 4.907185768843192e-12,\n",
       "      46: 3.2510105718586146e-11,\n",
       "      45: 1.755545708803652e-10,\n",
       "      44: 8.046251165350071e-10,\n",
       "      43: 3.2185004661400285e-09,\n",
       "      42: 1.1465907910623852e-08,\n",
       "      41: 3.6945703267565744e-08,\n",
       "      40: 1.0898982463931894e-07,\n",
       "      39: 2.9724497628905167e-07,\n",
       "      38: 7.554976480680063e-07,\n",
       "      37: 1.8015713146237074e-06,\n",
       "      36: 4.053535457903342e-06,\n",
       "      35: 8.647542310193795e-06,\n",
       "      34: 1.7565320317581147e-05,\n",
       "      33: 3.409738649883399e-05,\n",
       "      32: 6.345902487282993e-05,\n",
       "      31: 0.0001135582550355904,\n",
       "      30: 0.00019588798993639344,\n",
       "      29: 0.00032647998322732244,\n",
       "      28: 0.0005268199729349975,\n",
       "      27: 0.0008245877837243439,\n",
       "      26: 0.0012540605877474397,\n",
       "      25: 0.0018560096698662107,\n",
       "      24: 0.0026769370238454962,\n",
       "      23: 0.0037675409965232907,\n",
       "      22: 0.005180368870219524,\n",
       "      21: 0.006966702963398672,\n",
       "      20: 0.009172825568474917,\n",
       "      19: 0.011835903959322474,\n",
       "      18: 0.014979815948517508,\n",
       "      17: 0.01861128648149145,\n",
       "      16: 0.022716717322996918,\n",
       "      15: 0.027260060787596296,\n",
       "      14: 0.03218201620757896,\n",
       "      13: 0.03740072153853771,\n",
       "      12: 0.04281398386648397,\n",
       "      11: 0.04830295615705883,\n",
       "      10: 0.053737038724727945,\n",
       "      9: 0.05897967664909165,\n",
       "      8: 0.06389464970318263,\n",
       "      7: 0.0683524159615442,\n",
       "      6: 0.07223607595935921,\n",
       "      5: 0.07544656822421962,\n",
       "      4: 0.0779067824054442,\n",
       "      3: 0.07956437352045362,\n",
       "      2: 0.08039316907795838,\n",
       "      1: 0.08039316907795838})"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "other_card(50, 50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compute the expected value, `EV`, of this distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.958923738717876"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def EV(P): \n",
    "    \"\"\"Expected value of a probability distribution.\"\"\"\n",
    "    return sum(b * P[b] for b in P)\n",
    "\n",
    "EV(other_card(50, 50))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So on average, we expect about 8 drinks left on the other card.\n",
    "\n",
    "What if we were given two different gift cards to begin with? Say a 25- and a 50-drink card?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "25.008494650139617"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EV(other_card(25, 50))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's interesting. The expectation is almost exactly 25 drinks remaining on the other card (which in almost all cases will be the card that originally had 50 drinks). That doesn't mean there will always be 25 left, or even a number close to that. We see below that the remaining drinks on the other card is in the range 20-30 about half the time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5148807410710244"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = other_card(25, 50)\n",
    "sum(P[d] for d in range(20, 30))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can plot the probability distribution of `other_card(n, n)` for various values of  `n`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def show_other_cards(ns):\n",
    "    X = list(range(1, max(ns) // 2 + 1))\n",
    "    for n in ns:\n",
    "        P = other_card(n, n)\n",
    "        Y = [P[b] for b in X]\n",
    "        plt.plot(X, Y, '.-', label=f'n = {n}; EV = {float(EV(P)):.2f}')\n",
    "        plt.grid(True)\n",
    "        plt.xlabel('Drinks Remaining'); plt.ylabel('Probability')\n",
    "        plt.legend()\n",
    "        \n",
    "show_other_cards((20, 25, 30, 40, 50, 60, 75, 99))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Counting Multiplications in Matrix Multiplication\n",
    "\n",
    "> Given a sequence of matrices to be multiplied together,  what is the minimum number of multiplications of the constituent numbers, out of all the possible bracketings of the sequence?\n",
    "\n",
    "See [Wikipedia's matrix multiplication](https://en.wikipedia.org/wiki/Matrix_multiplication) page for a refresher if needed, but we will explain everything here. A matrix is a two-dimensional grid of numbers. When we multiply two matrices, we do a lot of multiplications of the constituent numbers.  In the diagram below we see that multiplying a matrix of dimensions $l × m$ by a $m × n$ matrix results in a $l × n$ matrix, $C$, where each element $C_{ij}$ is the dot product of two $m$ element vectors, for a total of $l × m × n$ multiplications of constituent numbers. (Forget about [Strassen](https://en.wikipedia.org/wiki/Strassen_algorithm) for now.)\n",
    "\n",
    "![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Matrix_multiplication_qtl1.svg/440px-Matrix_multiplication_qtl1.svg.png)\n",
    "\n",
    "$$\\sum_{k=1}^{m} A_{ik} × B_{kj} = C_{i,j}$$\n",
    "\n",
    "To multiply *three* matrices together, we could do either **(AB)C** or **A(BC)**; both give the same answer, but one might do fewer constituent multiplications. For a sequence of four matrices, there are five possible bracketings: **((AB)C)D,     (A(BC))D,     (AB)(CD),     A((BC)D)**, or     **A(B(CD))**. We want to find the bracketing with the fewest constituent multiplications.\n",
    "\n",
    "Here's how I start to think about it:\n",
    "\n",
    "- We don't need to represent the contents of the matrices, just the dimensions.\n",
    "- `Matrix(c, m, n)` will represent a matrix of dimensions $m × n$ that requires $c$ constituent multiplications. I'll implement `Matrix` as a `namedtuple` to which I add a method for multiplication. (To be clear: it doesn't do matrix multiplication; it just computed the count of constituent multiplications and the dimensions of the product.)\n",
    "- `M(m, n)` is a handy abbreviation for `Matrix(0, m, n)`\n",
    "\n",
    "Here is the  code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Matrix(namedtuple('_', 'count, d1, d2')):\n",
    "    \"\"\"Contains the dimensions (d1, d2) and a count of the number of multiplies.\"\"\"  \n",
    "    def __mul__(A, B):\n",
    "        assert A.d2 == B.d1\n",
    "        count = A.count + B.count + A.d1 * A.d2 * B.d2\n",
    "        return Matrix(count, A.d1, B.d2)\n",
    "\n",
    "def M(m, n): return Matrix(0, m, n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some examples of use:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix(count=5000, d1=10, d2=5)"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A, B, C = M(10, 100), M(100, 5), M(5, 20)\n",
    "\n",
    "A * B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix(count=10000, d1=100, d2=20)"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B * C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix(count=6000, d1=10, d2=20)"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A * B) * C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix(count=30000, d1=10, d2=20)"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * (B * C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The bracketing **(AB)C** results in 6000 multiplications; five times less than **A(BC)**. Here's how to find the best product of a sequence of matrices:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- `multbest(matrices)` will return a `Matrix` that is the product of `matrices` and has the minimum number of constituent multiplications.\n",
    "- The **brute force enumeration** approach  would be to try every possible bracketing. How many bracketings are there? It turns out the answer is again the [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number), so for 10 matrices there are only 4,862 possibilities; no problem. But for 25 matrices, there are 1,289,904,147,324 possibilities.\n",
    "- We don't need to evaluate every bracketing. Consider splitting the sequence of matrices into two parts, *left* and *right*. We don't need to find every way of bracketing *left* and combine each of those with every way of bracketing *right*; we only need to find the one best way of bracketing *left* and combine it with the one best way of bracketing *right*. We do have to consider every way of splitting *left* and *right*, but there are only *O(n)* of those.\n",
    "- So the algorithm for `multbest` will be a form of the **incremental enumeration** with **remembering** approach:\n",
    "   - For every way of splitting the sequence of matrices, recursively find the best matrix products for the *left* and *right* parts of the split, and multiply them to get a candidate.\n",
    "   - Out of those candidates, return the one with the minimum number of multiplications. (Since `Matrix` was defined as a namedtuple with the count as the first field, the `min` matrix is the one with the minimum count.)\n",
    "   - We use `@lru_cache` to stop the algorithm from repeating itself."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [],
   "source": [
    "@lru_cache(None)\n",
    "def multbest(matrices):\n",
    "    \"\"\"Find the product of matrices that uses the least number of multiplications.\"\"\"\n",
    "    return (matrices[0] if len(matrices) == 1 else\n",
    "            min(multbest(left) * multbest(right) \n",
    "                for left, right in splits(matrices)))\n",
    "    \n",
    "def splits(seq): return [(seq[:i], seq[i:]) for i in range(1, len(seq))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('A', 'BCDEF'),\n",
       " ('AB', 'CDEF'),\n",
       " ('ABC', 'DEF'),\n",
       " ('ABCD', 'EF'),\n",
       " ('ABCDE', 'F')]"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "splits(\"ABCDEF\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix(count=6000, d1=10, d2=20)"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multbest((A, B, C))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix(count=75350, d1=10, d2=20)"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "example = (A, B, C, M(20, 99), M(99, 10), M(10, 100), M(100, 80), M(80, 10), A, B, C)\n",
    "\n",
    "multbest(example)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see it takes 75,350 constituent multiplications to work out the matrix product of the sequence of eleven `example` matrices. Is that a lot or a little? We can compare this optimal result to the pessimal result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [],
   "source": [
    "def multworst(matrices):\n",
    "    \"\"\"Find the product of matrices that uses the MOST number of multiplications.\"\"\"\n",
    "    return (matrices[0] if len(matrices) == 1 else\n",
    "            max(multworst(left) * multworst(right) \n",
    "                for left, right in splits(matrices)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix(count=3407000, d1=10, d2=20)"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multworst(example)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's 45 times more work! It seems worthwhile to take the optimal approach."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conclusion\n",
    "\n",
    "Thanks for making it all the way to the end of the notebook. I hope you've learned something about methods for counting things, and that you can apply them to your own problems."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}
